# Properties

 Label 1200.2.v.m Level $1200$ Weight $2$ Character orbit 1200.v Analytic conductor $9.582$ Analytic rank $0$ Dimension $16$ 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: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: 16.0.6040479020157644046336.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561$$ x^16 - 7*x^12 - 32*x^8 - 567*x^4 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 600) 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{3} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{7} + (\beta_{11} - \beta_{3}) q^{9}+O(q^{10})$$ q + b9 * q^3 + (-b5 - b2 - b1) * q^7 + (b11 - b3) * q^9 $$q + \beta_{9} q^{3} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{7} + (\beta_{11} - \beta_{3}) q^{9} + (\beta_{10} + \beta_{8}) q^{11} + ( - \beta_{14} + 3 \beta_{13} - 3 \beta_{9}) q^{13} + ( - \beta_{12} + \beta_{5} - \beta_1) q^{17} + ( - 2 \beta_{11} + 2 \beta_{4} + \beta_{3}) q^{19} + (\beta_{10} + \beta_{8} + 1) q^{21} + ( - \beta_{15} + 2 \beta_{13} + 2 \beta_{9}) q^{23} + ( - \beta_{12} - 2 \beta_{5} + \cdots - 2 \beta_1) q^{27}+ \cdots + (\beta_{11} + 3 \beta_{6} + \cdots - 4 \beta_{3}) q^{99}+O(q^{100})$$ q + b9 * q^3 + (-b5 - b2 - b1) * q^7 + (b11 - b3) * q^9 + (b10 + b8) * q^11 + (-b14 + 3*b13 - 3*b9) * q^13 + (-b12 + b5 - b1) * q^17 + (-2*b11 + 2*b4 + b3) * q^19 + (b10 + b8 + 1) * q^21 + (-b15 + 2*b13 + 2*b9) * q^23 + (-b12 - 2*b5 - b2 - 2*b1) * q^27 + (b11 + 3*b6 - 2*b4) * q^29 + (-b10 - b7 + 3) * q^31 + (-3*b14 + 3*b13 - 2*b9) * q^33 + (2*b5 + 2*b1) * q^37 + (-2*b11 + b6 - b4 - 5*b3) * q^39 + (b10 + 3*b8 + 2*b7) * q^41 + (5*b14 - b13 + b9) * q^43 + 2*b12 * q^47 + (b11 - b4 + 4*b3) * q^49 + (-2*b10 + b8 + 4) * q^51 + (b15 + 2*b13 + 2*b9) * q^53 + (2*b12 + b5 + 2*b2 - 2*b1) * q^57 + (2*b11 + 2*b4) * q^59 + (-b10 - b7 - 1) * q^61 + (-3*b14 + 3*b13 - b9) * q^63 - b2 * q^67 + (b11 + b6 + 2*b4 - 8*b3) * q^69 + (2*b10 + 3*b8 + b7) * q^71 + (6*b14 - b13 + b9) * q^73 + (-b12 - 2*b5 + 2*b1) * q^77 + (2*b11 - 2*b4 + 4*b3) * q^79 + (-b10 + 2*b8 - 2*b7 + 3) * q^81 + (2*b15 + b13 + b9) * q^83 + (-b12 + 4*b5 + 8*b2 + 4*b1) * q^87 + (-3*b11 - 2*b6 - b4) * q^89 + (-b10 - b7 - 7) * q^91 + (-b15 - b14 - b13 + 3*b9) * q^93 + (-5*b5 - 3*b2 - 5*b1) * q^97 + (b11 + 3*b6 - 3*b4 - 4*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q+O(q^{10})$$ 16 * q $$16 q + 16 q^{21} + 40 q^{31} + 52 q^{51} - 24 q^{61} + 28 q^{81} - 120 q^{91}+O(q^{100})$$ 16 * q + 16 * q^21 + 40 * q^31 + 52 * q^51 - 24 * q^61 + 28 * q^81 - 120 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -11\nu^{13} - 4\nu^{9} - 296\nu^{5} + 6885\nu ) / 4860$$ (-11*v^13 - 4*v^9 - 296*v^5 + 6885*v) / 4860 $$\beta_{3}$$ $$=$$ $$( -7\nu^{14} - 32\nu^{10} + 62\nu^{6} + 6561\nu^{2} ) / 7290$$ (-7*v^14 - 32*v^10 + 62*v^6 + 6561*v^2) / 7290 $$\beta_{4}$$ $$=$$ $$( 7\nu^{14} + 32\nu^{10} - 62\nu^{6} + 729\nu^{2} ) / 7290$$ (7*v^14 + 32*v^10 - 62*v^6 + 729*v^2) / 7290 $$\beta_{5}$$ $$=$$ $$( 7\nu^{13} + 32\nu^{9} - 62\nu^{5} - 6561\nu ) / 2430$$ (7*v^13 + 32*v^9 - 62*v^5 - 6561*v) / 2430 $$\beta_{6}$$ $$=$$ $$( 19\nu^{14} - 52\nu^{10} + 1012\nu^{6} - 7533\nu^{2} ) / 14580$$ (19*v^14 - 52*v^10 + 1012*v^6 - 7533*v^2) / 14580 $$\beta_{7}$$ $$=$$ $$( -7\nu^{12} - 32\nu^{8} + 62\nu^{4} + 5751 ) / 810$$ (-7*v^12 - 32*v^8 + 62*v^4 + 5751) / 810 $$\beta_{8}$$ $$=$$ $$( -5\nu^{12} + 8\nu^{8} + 52\nu^{4} + 2943 ) / 540$$ (-5*v^12 + 8*v^8 + 52*v^4 + 2943) / 540 $$\beta_{9}$$ $$=$$ $$( \nu^{15} - 7\nu^{11} - 32\nu^{7} - 567\nu^{3} ) / 2187$$ (v^15 - 7*v^11 - 32*v^7 - 567*v^3) / 2187 $$\beta_{10}$$ $$=$$ $$( \nu^{12} + 2\nu^{8} - 32\nu^{4} - 657 ) / 90$$ (v^12 + 2*v^8 - 32*v^4 - 657) / 90 $$\beta_{11}$$ $$=$$ $$( -17\nu^{14} + 38\nu^{10} + 382\nu^{6} + 12231\nu^{2} ) / 7290$$ (-17*v^14 + 38*v^10 + 382*v^6 + 12231*v^2) / 7290 $$\beta_{12}$$ $$=$$ $$( -37\nu^{13} + 16\nu^{9} + 1184\nu^{5} + 20979\nu ) / 4860$$ (-37*v^13 + 16*v^9 + 1184*v^5 + 20979*v) / 4860 $$\beta_{13}$$ $$=$$ $$( -7\nu^{15} - 32\nu^{11} + 62\nu^{7} + 6561\nu^{3} ) / 7290$$ (-7*v^15 - 32*v^11 + 62*v^7 + 6561*v^3) / 7290 $$\beta_{14}$$ $$=$$ $$( -43\nu^{15} - 104\nu^{11} + 2024\nu^{7} + 28593\nu^{3} ) / 43740$$ (-43*v^15 - 104*v^11 + 2024*v^7 + 28593*v^3) / 43740 $$\beta_{15}$$ $$=$$ $$( \nu^{15} - 359\nu^{3} ) / 540$$ (v^15 - 359*v^3) / 540
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3}$$ b4 + b3 $$\nu^{3}$$ $$=$$ $$\beta_{15} - \beta_{14} + 2\beta_{13} - 2\beta_{9}$$ b15 - b14 + 2*b13 - 2*b9 $$\nu^{4}$$ $$=$$ $$-4\beta_{10} - 2\beta_{8} - 3\beta_{7} + 3$$ -4*b10 - 2*b8 - 3*b7 + 3 $$\nu^{5}$$ $$=$$ $$2\beta_{12} - \beta_{5} - 8\beta_{2}$$ 2*b12 - b5 - 8*b2 $$\nu^{6}$$ $$=$$ $$6\beta_{11} + 10\beta_{6} - 4\beta_{4} - 5\beta_{3}$$ 6*b11 + 10*b6 - 4*b4 - 5*b3 $$\nu^{7}$$ $$=$$ $$6\beta_{15} + 24\beta_{14} - 13\beta_{13}$$ 6*b15 + 24*b14 - 13*b13 $$\nu^{8}$$ $$=$$ $$\beta_{10} + 18\beta_{8} - 18\beta_{7} + 37$$ b10 + 18*b8 - 18*b7 + 37 $$\nu^{9}$$ $$=$$ $$17\beta_{12} + 74\beta_{5} + 37\beta_{2} + 74\beta_1$$ 17*b12 + 74*b5 + 37*b2 + 74*b1 $$\nu^{10}$$ $$=$$ $$51\beta_{11} - 20\beta_{6} + 40\beta_{4} - 111\beta_{3}$$ 51*b11 - 20*b6 + 40*b4 - 111*b3 $$\nu^{11}$$ $$=$$ $$20\beta_{15} - 80\beta_{14} - 253\beta_{9}$$ 20*b15 - 80*b14 - 253*b9 $$\nu^{12}$$ $$=$$ $$-40\beta_{10} - 100\beta_{8} - 60\beta_{7} + 679$$ -40*b10 - 100*b8 - 60*b7 + 679 $$\nu^{13}$$ $$=$$ $$-60\beta_{12} - 240\beta_{2} + 599\beta_1$$ -60*b12 - 240*b2 + 599*b1 $$\nu^{14}$$ $$=$$ $$-180\beta_{11} + 180\beta_{6} + 719\beta_{4} + 359\beta_{3}$$ -180*b11 + 180*b6 + 719*b4 + 359*b3 $$\nu^{15}$$ $$=$$ $$899\beta_{15} - 359\beta_{14} + 718\beta_{13} - 718\beta_{9}$$ 899*b15 - 359*b14 + 718*b13 - 718*b9

## Character values

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

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-\beta_{3}$$ $$1$$ $$1$$

## 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
 1.73122 + 0.0537601i 1.47240 − 0.912166i 0.912166 − 1.47240i 0.0537601 + 1.73122i −0.0537601 − 1.73122i −0.912166 + 1.47240i −1.47240 + 0.912166i −1.73122 − 0.0537601i 1.73122 − 0.0537601i 1.47240 + 0.912166i 0.912166 + 1.47240i 0.0537601 − 1.73122i −0.0537601 + 1.73122i −0.912166 − 1.47240i −1.47240 − 0.912166i −1.73122 + 0.0537601i
0 −1.73122 + 0.0537601i 0 0 0 −0.560232 0.560232i 0 2.99422 0.186141i 0
257.2 0 −1.47240 0.912166i 0 0 0 −1.78498 1.78498i 0 1.33591 + 2.68614i 0
257.3 0 −0.912166 1.47240i 0 0 0 1.78498 + 1.78498i 0 −1.33591 + 2.68614i 0
257.4 0 −0.0537601 + 1.73122i 0 0 0 −0.560232 0.560232i 0 −2.99422 0.186141i 0
257.5 0 0.0537601 1.73122i 0 0 0 0.560232 + 0.560232i 0 −2.99422 0.186141i 0
257.6 0 0.912166 + 1.47240i 0 0 0 −1.78498 1.78498i 0 −1.33591 + 2.68614i 0
257.7 0 1.47240 + 0.912166i 0 0 0 1.78498 + 1.78498i 0 1.33591 + 2.68614i 0
257.8 0 1.73122 0.0537601i 0 0 0 0.560232 + 0.560232i 0 2.99422 0.186141i 0
593.1 0 −1.73122 0.0537601i 0 0 0 −0.560232 + 0.560232i 0 2.99422 + 0.186141i 0
593.2 0 −1.47240 + 0.912166i 0 0 0 −1.78498 + 1.78498i 0 1.33591 2.68614i 0
593.3 0 −0.912166 + 1.47240i 0 0 0 1.78498 1.78498i 0 −1.33591 2.68614i 0
593.4 0 −0.0537601 1.73122i 0 0 0 −0.560232 + 0.560232i 0 −2.99422 + 0.186141i 0
593.5 0 0.0537601 + 1.73122i 0 0 0 0.560232 0.560232i 0 −2.99422 + 0.186141i 0
593.6 0 0.912166 1.47240i 0 0 0 −1.78498 + 1.78498i 0 −1.33591 2.68614i 0
593.7 0 1.47240 0.912166i 0 0 0 1.78498 1.78498i 0 1.33591 2.68614i 0
593.8 0 1.73122 + 0.0537601i 0 0 0 0.560232 0.560232i 0 2.99422 + 0.186141i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.8 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.m 16
3.b odd 2 1 inner 1200.2.v.m 16
4.b odd 2 1 600.2.r.f 16
5.b even 2 1 inner 1200.2.v.m 16
5.c odd 4 2 inner 1200.2.v.m 16
12.b even 2 1 600.2.r.f 16
15.d odd 2 1 inner 1200.2.v.m 16
15.e even 4 2 inner 1200.2.v.m 16
20.d odd 2 1 600.2.r.f 16
20.e even 4 2 600.2.r.f 16
60.h even 2 1 600.2.r.f 16
60.l odd 4 2 600.2.r.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.r.f 16 4.b odd 2 1
600.2.r.f 16 12.b even 2 1
600.2.r.f 16 20.d odd 2 1
600.2.r.f 16 20.e even 4 2
600.2.r.f 16 60.h even 2 1
600.2.r.f 16 60.l odd 4 2
1200.2.v.m 16 1.a even 1 1 trivial
1200.2.v.m 16 3.b odd 2 1 inner
1200.2.v.m 16 5.b even 2 1 inner
1200.2.v.m 16 5.c odd 4 2 inner
1200.2.v.m 16 15.d odd 2 1 inner
1200.2.v.m 16 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 $$S_{2}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{8} + 41T_{7}^{4} + 16$$ T7^8 + 41*T7^4 + 16 $$T_{11}^{4} + 19T_{11}^{2} + 16$$ T11^4 + 19*T11^2 + 16 $$T_{17}^{8} + 1649T_{17}^{4} + 256$$ T17^8 + 1649*T17^4 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} - 7 T^{12} + \cdots + 6561$$
$5$ $$T^{16}$$
$7$ $$(T^{8} + 41 T^{4} + 16)^{2}$$
$11$ $$(T^{4} + 19 T^{2} + 16)^{4}$$
$13$ $$(T^{8} + 1449 T^{4} + 331776)^{2}$$
$17$ $$(T^{8} + 1649 T^{4} + 256)^{2}$$
$19$ $$(T^{4} + 74 T^{2} + 841)^{4}$$
$23$ $$(T^{8} + 5904 T^{4} + 331776)^{2}$$
$29$ $$(T^{4} - 112 T^{2} + 1024)^{4}$$
$31$ $$(T^{2} - 5 T - 2)^{8}$$
$37$ $$(T^{8} + 656 T^{4} + 4096)^{2}$$
$41$ $$(T^{4} + 139 T^{2} + 4624)^{4}$$
$43$ $$(T^{8} + 9401 T^{4} + 11316496)^{2}$$
$47$ $$(T^{4} + 4096)^{4}$$
$53$ $$(T^{8} + 13328 T^{4} + 4096)^{2}$$
$59$ $$(T^{4} - 172 T^{2} + 4096)^{4}$$
$61$ $$(T^{2} + 3 T - 6)^{8}$$
$67$ $$(T^{4} + 9)^{4}$$
$71$ $$(T^{4} + 112 T^{2} + 1024)^{4}$$
$73$ $$(T^{8} + 19481 T^{4} + 59969536)^{2}$$
$79$ $$(T^{4} + 84 T^{2} + 576)^{4}$$
$83$ $$(T^{8} + 17729 T^{4} + 16777216)^{2}$$
$89$ $$(T^{4} - 259 T^{2} + 64)^{4}$$
$97$ $$(T^{8} + 10073 T^{4} + 21381376)^{2}$$