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Properties

Label 1200.3.j.d
Level $1200$
Weight $3$
Character orbit 1200.j
Analytic conductor $32.698$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(799,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.799");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.j (of order \(2\), degree \(1\), 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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{5} - \beta_1) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{5} - \beta_1) q^{7} + 3 q^{9} - \beta_{4} q^{11} - \beta_{2} q^{13} + ( - \beta_{6} - 12 \beta_{2}) q^{17} + ( - \beta_{4} + \beta_{3}) q^{19} + ( - \beta_{7} - 3) q^{21} + ( - \beta_{5} + 12 \beta_1) q^{23} + 3 \beta_1 q^{27} + ( - \beta_{7} + 24) q^{29} + (\beta_{4} - 15 \beta_{3}) q^{31} + \beta_{6} q^{33} + (2 \beta_{6} - 22 \beta_{2}) q^{37} - \beta_{3} q^{39} + (3 \beta_{7} - 6) q^{41} + (\beta_{5} + 21 \beta_1) q^{43} + ( - 4 \beta_{5} - 6 \beta_1) q^{47} + (2 \beta_{7} + 86) q^{49} + (3 \beta_{4} - 12 \beta_{3}) q^{51} + ( - 3 \beta_{6} - 30 \beta_{2}) q^{53} + (\beta_{6} + 3 \beta_{2}) q^{57} + (4 \beta_{4} + 30 \beta_{3}) q^{59} + ( - 2 \beta_{7} - 37) q^{61} + (3 \beta_{5} - 3 \beta_1) q^{63} + ( - 3 \beta_{5} + 43 \beta_1) q^{67} + (\beta_{7} + 36) q^{69} + ( - \beta_{4} - 6 \beta_{3}) q^{71} + (2 \beta_{6} - 70 \beta_{2}) q^{73} + ( - \beta_{6} - 132 \beta_{2}) q^{77} + (4 \beta_{4} + 28 \beta_{3}) q^{79} + 9 q^{81} + 42 \beta_1 q^{83} + (3 \beta_{5} + 24 \beta_1) q^{87} + 96 q^{89} + ( - \beta_{4} + \beta_{3}) q^{91} + ( - \beta_{6} - 45 \beta_{2}) q^{93} + ( - 4 \beta_{6} - 35 \beta_{2}) q^{97} - 3 \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} - 24 q^{21} + 192 q^{29} - 48 q^{41} + 688 q^{49} - 296 q^{61} + 288 q^{69} + 72 q^{81} + 768 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 2\nu^{3} - 9\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} - 16\nu^{5} - 8\nu^{3} - 81\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 153 ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} + 64\nu^{5} + 248\nu^{3} + 1071\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{6} - 20\nu^{4} - 28\nu^{2} - 90 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} - 15 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} - 16\nu^{5} - 62\nu^{3} - 81\nu ) / 9 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{4} - 6\beta_{2} - 6\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 3\beta_{5} - 30\beta_{3} - 30 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 24\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{6} - 15\beta_{5} + 42\beta_{3} - 42 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 3\beta_{4} - 186\beta_{2} + 186\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{6} - 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 39\beta_{4} - 498\beta_{2} - 498\beta_1 ) / 24 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
−0.396143 + 1.68614i
−0.396143 − 1.68614i
1.26217 + 1.18614i
1.26217 − 1.18614i
0.396143 + 1.68614i
0.396143 − 1.68614i
−1.26217 + 1.18614i
−1.26217 − 1.18614i
0 −1.73205 0 0 0 −9.75707 0 3.00000 0
799.2 0 −1.73205 0 0 0 −9.75707 0 3.00000 0
799.3 0 −1.73205 0 0 0 13.2212 0 3.00000 0
799.4 0 −1.73205 0 0 0 13.2212 0 3.00000 0
799.5 0 1.73205 0 0 0 −13.2212 0 3.00000 0
799.6 0 1.73205 0 0 0 −13.2212 0 3.00000 0
799.7 0 1.73205 0 0 0 9.75707 0 3.00000 0
799.8 0 1.73205 0 0 0 9.75707 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.j.d 8
3.b odd 2 1 3600.3.j.j 8
4.b odd 2 1 inner 1200.3.j.d 8
5.b even 2 1 inner 1200.3.j.d 8
5.c odd 4 1 1200.3.e.k ✓ 4
5.c odd 4 1 1200.3.e.m yes 4
12.b even 2 1 3600.3.j.j 8
15.d odd 2 1 3600.3.j.j 8
15.e even 4 1 3600.3.e.bc 4
15.e even 4 1 3600.3.e.bf 4
20.d odd 2 1 inner 1200.3.j.d 8
20.e even 4 1 1200.3.e.k ✓ 4
20.e even 4 1 1200.3.e.m yes 4
60.h even 2 1 3600.3.j.j 8
60.l odd 4 1 3600.3.e.bc 4
60.l odd 4 1 3600.3.e.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.3.e.k ✓ 4 5.c odd 4 1
1200.3.e.k ✓ 4 20.e even 4 1
1200.3.e.m yes 4 5.c odd 4 1
1200.3.e.m yes 4 20.e even 4 1
1200.3.j.d 8 1.a even 1 1 trivial
1200.3.j.d 8 4.b odd 2 1 inner
1200.3.j.d 8 5.b even 2 1 inner
1200.3.j.d 8 20.d odd 2 1 inner
3600.3.e.bc 4 15.e even 4 1
3600.3.e.bc 4 60.l odd 4 1
3600.3.e.bf 4 15.e even 4 1
3600.3.e.bf 4 60.l odd 4 1
3600.3.j.j 8 3.b odd 2 1
3600.3.j.j 8 12.b even 2 1
3600.3.j.j 8 15.d odd 2 1
3600.3.j.j 8 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{4} - 270T_{7}^{2} + 16641 \) Copy content Toggle raw display
\( T_{13}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 270 T^{2} + 16641)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 132)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1080 T^{2} + 63504)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 270 T^{2} + 16641)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 1128 T^{2} + 90000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 48 T + 180)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1614 T^{2} + 294849)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4136 T^{2} + 1210000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T - 3528)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2910 T^{2} + 1418481)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4440 T^{2} + 4016016)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8928 T^{2} + 7096896)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 9624 T^{2} + 345744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 74 T - 215)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 13470 T^{2} + 19000881)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 480 T^{2} + 576)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 12968 T^{2} + 10995856)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 8928 T^{2} + 57600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 5292)^{4} \) Copy content Toggle raw display
$89$ \( (T - 96)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 15122 T^{2} + 26122321)^{2} \) Copy content Toggle raw display
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