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Properties

Label 1200.3.e.k
Level $1200$
Weight $3$
Character orbit 1200.e
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(751,1200)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.751");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.e (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{2} + \beta_1) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{2} + \beta_1) q^{7} - 3 q^{9} - \beta_1 q^{11} - q^{13} + (\beta_{3} + 12) q^{17} + (\beta_{2} + \beta_1) q^{19} + ( - \beta_{3} - 3) q^{21} + (12 \beta_{2} + \beta_1) q^{23} - 3 \beta_{2} q^{27} + (\beta_{3} - 24) q^{29} + (15 \beta_{2} + \beta_1) q^{31} + \beta_{3} q^{33} + ( - 2 \beta_{3} + 22) q^{37} - \beta_{2} q^{39} + (3 \beta_{3} - 6) q^{41} + (21 \beta_{2} - \beta_1) q^{43} + (6 \beta_{2} - 4 \beta_1) q^{47} + ( - 2 \beta_{3} - 86) q^{49} + (12 \beta_{2} + 3 \beta_1) q^{51} + ( - 3 \beta_{3} - 30) q^{53} + ( - \beta_{3} - 3) q^{57} + (30 \beta_{2} - 4 \beta_1) q^{59} + ( - 2 \beta_{3} - 37) q^{61} + ( - 3 \beta_{2} - 3 \beta_1) q^{63} + ( - 43 \beta_{2} - 3 \beta_1) q^{67} + ( - \beta_{3} - 36) q^{69} + (6 \beta_{2} - \beta_1) q^{71} + (2 \beta_{3} - 70) q^{73} + (\beta_{3} + 132) q^{77} + (28 \beta_{2} - 4 \beta_1) q^{79} + 9 q^{81} + 42 \beta_{2} q^{83} + ( - 24 \beta_{2} + 3 \beta_1) q^{87} - 96 q^{89} + ( - \beta_{2} - \beta_1) q^{91} + ( - \beta_{3} - 45) q^{93} + (4 \beta_{3} + 35) q^{97} + 3 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} - 4 q^{13} + 48 q^{17} - 12 q^{21} - 96 q^{29} + 88 q^{37} - 24 q^{41} - 344 q^{49} - 120 q^{53} - 12 q^{57} - 148 q^{61} - 144 q^{69} - 280 q^{73} + 528 q^{77} + 36 q^{81} - 384 q^{89} - 180 q^{93} + 140 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 44\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} + 11 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{3} ) / 11 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11\beta_{2} - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{3} ) / 6 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
1.65831 − 2.87228i
−1.65831 + 2.87228i
−1.65831 − 2.87228i
1.65831 + 2.87228i
0 − 1.73205i 0 0 0 − 13.2212i 0 −3.00000 0
751.2 0 − 1.73205i 0 0 0 9.75707i 0 −3.00000 0
751.3 0 1.73205i 0 0 0 − 9.75707i 0 −3.00000 0
751.4 0 1.73205i 0 0 0 13.2212i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

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
\( T_{11}^{2} + 132 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 270 T^{2} + 16641 \) Copy content Toggle raw display
$11$ \( (T^{2} + 132)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 24 T - 252)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 270 T^{2} + 16641 \) Copy content Toggle raw display
$23$ \( T^{4} + 1128 T^{2} + 90000 \) Copy content Toggle raw display
$29$ \( (T^{2} + 48 T + 180)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 1614 T^{2} + 294849 \) Copy content Toggle raw display
$37$ \( (T^{2} - 44 T - 1100)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T - 3528)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 2910 T^{2} + 1418481 \) Copy content Toggle raw display
$47$ \( T^{4} + 4440 T^{2} + 4016016 \) Copy content Toggle raw display
$53$ \( (T^{2} + 60 T - 2664)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 9624 T^{2} + 345744 \) Copy content Toggle raw display
$61$ \( (T^{2} + 74 T - 215)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 13470 T^{2} + 19000881 \) Copy content Toggle raw display
$71$ \( T^{4} + 480T^{2} + 576 \) Copy content Toggle raw display
$73$ \( (T^{2} + 140 T + 3316)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 8928 T^{2} + 57600 \) Copy content Toggle raw display
$83$ \( (T^{2} + 5292)^{2} \) Copy content Toggle raw display
$89$ \( (T + 96)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 70 T - 5111)^{2} \) Copy content Toggle raw display
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