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

Downloads

Learn more

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: \( x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{7} + (\beta_{11} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{3} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{7} + (\beta_{11} - \beta_{3}) q^{9} + (\beta_{10} + \beta_{8}) q^{11} + ( - \beta_{14} + 3 \beta_{13} - 3 \beta_{9}) q^{13} + ( - \beta_{12} + \beta_{5} - \beta_1) q^{17} + ( - 2 \beta_{11} + 2 \beta_{4} + \beta_{3}) q^{19} + (\beta_{10} + \beta_{8} + 1) q^{21} + ( - \beta_{15} + 2 \beta_{13} + 2 \beta_{9}) q^{23} + ( - \beta_{12} - 2 \beta_{5} + \cdots - 2 \beta_1) q^{27}+ \cdots + (\beta_{11} + 3 \beta_{6} + \cdots - 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{21} + 40 q^{31} + 52 q^{51} - 24 q^{61} + 28 q^{81} - 120 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{13} - 4\nu^{9} - 296\nu^{5} + 6885\nu ) / 4860 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{14} - 32\nu^{10} + 62\nu^{6} + 6561\nu^{2} ) / 7290 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{14} + 32\nu^{10} - 62\nu^{6} + 729\nu^{2} ) / 7290 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{13} + 32\nu^{9} - 62\nu^{5} - 6561\nu ) / 2430 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{14} - 52\nu^{10} + 1012\nu^{6} - 7533\nu^{2} ) / 14580 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{12} - 32\nu^{8} + 62\nu^{4} + 5751 ) / 810 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{12} + 8\nu^{8} + 52\nu^{4} + 2943 ) / 540 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} - 7\nu^{11} - 32\nu^{7} - 567\nu^{3} ) / 2187 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{12} + 2\nu^{8} - 32\nu^{4} - 657 ) / 90 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -17\nu^{14} + 38\nu^{10} + 382\nu^{6} + 12231\nu^{2} ) / 7290 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -37\nu^{13} + 16\nu^{9} + 1184\nu^{5} + 20979\nu ) / 4860 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -7\nu^{15} - 32\nu^{11} + 62\nu^{7} + 6561\nu^{3} ) / 7290 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -43\nu^{15} - 104\nu^{11} + 2024\nu^{7} + 28593\nu^{3} ) / 43740 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} - 359\nu^{3} ) / 540 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + 2\beta_{13} - 2\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{10} - 2\beta_{8} - 3\beta_{7} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{12} - \beta_{5} - 8\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{11} + 10\beta_{6} - 4\beta_{4} - 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6\beta_{15} + 24\beta_{14} - 13\beta_{13} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{10} + 18\beta_{8} - 18\beta_{7} + 37 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 17\beta_{12} + 74\beta_{5} + 37\beta_{2} + 74\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 51\beta_{11} - 20\beta_{6} + 40\beta_{4} - 111\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 20\beta_{15} - 80\beta_{14} - 253\beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -40\beta_{10} - 100\beta_{8} - 60\beta_{7} + 679 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -60\beta_{12} - 240\beta_{2} + 599\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -180\beta_{11} + 180\beta_{6} + 719\beta_{4} + 359\beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 899\beta_{15} - 359\beta_{14} + 718\beta_{13} - 718\beta_{9} \) Copy content Toggle raw display

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:

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 \(S_{2}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{8} + 41T_{7}^{4} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} + 19T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{8} + 1649T_{17}^{4} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

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