Properties

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

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(401,1200)]
 
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([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (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: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
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 \(\beta = \frac{1}{2}(1 + \sqrt{-35})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 8 q^{7} + (\beta - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 8 q^{7} + (\beta - 9) q^{9} + ( - 6 \beta + 3) q^{11} - 2 q^{13} + (6 \beta - 3) q^{17} - 11 q^{19} + 8 \beta q^{21} + (12 \beta - 6) q^{23} + ( - 8 \beta - 9) q^{27} + (12 \beta - 6) q^{29} + 46 q^{31} + ( - 3 \beta + 54) q^{33} + 16 q^{37} - 2 \beta q^{39} + (18 \beta - 9) q^{41} + 62 q^{43} + ( - 12 \beta + 6) q^{47} + 15 q^{49} + (3 \beta - 54) q^{51} + (12 \beta - 6) q^{53} - 11 \beta q^{57} + (24 \beta - 12) q^{59} - 16 q^{61} + (8 \beta - 72) q^{63} + 113 q^{67} + (6 \beta - 108) q^{69} + (36 \beta - 18) q^{71} - 101 q^{73} + ( - 48 \beta + 24) q^{77} - 68 q^{79} + ( - 17 \beta + 72) q^{81} + ( - 6 \beta + 3) q^{83} + (6 \beta - 108) q^{87} + (18 \beta - 9) q^{89} - 16 q^{91} + 46 \beta q^{93} + 22 q^{97} + (51 \beta + 27) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 16 q^{7} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 16 q^{7} - 17 q^{9} - 4 q^{13} - 22 q^{19} + 8 q^{21} - 26 q^{27} + 92 q^{31} + 105 q^{33} + 32 q^{37} - 2 q^{39} + 124 q^{43} + 30 q^{49} - 105 q^{51} - 11 q^{57} - 32 q^{61} - 136 q^{63} + 226 q^{67} - 210 q^{69} - 202 q^{73} - 136 q^{79} + 127 q^{81} - 210 q^{87} - 32 q^{91} + 46 q^{93} + 44 q^{97} + 105 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
0.500000 2.95804i
0.500000 + 2.95804i
0 0.500000 2.95804i 0 0 0 8.00000 0 −8.50000 2.95804i 0
401.2 0 0.500000 + 2.95804i 0 0 0 8.00000 0 −8.50000 + 2.95804i 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
3.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.l.m 2
3.b odd 2 1 inner 1200.3.l.m 2
4.b odd 2 1 300.3.g.f 2
5.b even 2 1 1200.3.l.k 2
5.c odd 4 2 1200.3.c.j 4
12.b even 2 1 300.3.g.f 2
15.d odd 2 1 1200.3.l.k 2
15.e even 4 2 1200.3.c.j 4
20.d odd 2 1 300.3.g.g yes 2
20.e even 4 2 300.3.b.d 4
60.h even 2 1 300.3.g.g yes 2
60.l odd 4 2 300.3.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.d 4 20.e even 4 2
300.3.b.d 4 60.l odd 4 2
300.3.g.f 2 4.b odd 2 1
300.3.g.f 2 12.b even 2 1
300.3.g.g yes 2 20.d odd 2 1
300.3.g.g yes 2 60.h even 2 1
1200.3.c.j 4 5.c odd 4 2
1200.3.c.j 4 15.e even 4 2
1200.3.l.k 2 5.b even 2 1
1200.3.l.k 2 15.d odd 2 1
1200.3.l.m 2 1.a even 1 1 trivial
1200.3.l.m 2 3.b odd 2 1 inner

Hecke kernels

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

\( T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 315 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 315 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 315 \) Copy content Toggle raw display
$19$ \( (T + 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1260 \) Copy content Toggle raw display
$29$ \( T^{2} + 1260 \) Copy content Toggle raw display
$31$ \( (T - 46)^{2} \) Copy content Toggle raw display
$37$ \( (T - 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2835 \) Copy content Toggle raw display
$43$ \( (T - 62)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1260 \) Copy content Toggle raw display
$53$ \( T^{2} + 1260 \) Copy content Toggle raw display
$59$ \( T^{2} + 5040 \) Copy content Toggle raw display
$61$ \( (T + 16)^{2} \) Copy content Toggle raw display
$67$ \( (T - 113)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 11340 \) Copy content Toggle raw display
$73$ \( (T + 101)^{2} \) Copy content Toggle raw display
$79$ \( (T + 68)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 315 \) Copy content Toggle raw display
$89$ \( T^{2} + 2835 \) Copy content Toggle raw display
$97$ \( (T - 22)^{2} \) Copy content Toggle raw display
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