Properties

Label 1200.4.a.bq
Level $1200$
Weight $4$
Character orbit 1200.a
Self dual yes
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(1,1200)]
 
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, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

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: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{129}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - 3 \beta - 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + ( - 3 \beta - 1) q^{7} + 9 q^{9} + (\beta - 37) q^{11} + (\beta - 49) q^{13} + (5 \beta + 39) q^{17} + ( - 8 \beta + 40) q^{19} + ( - 9 \beta - 3) q^{21} + (8 \beta + 20) q^{23} + 27 q^{27} + ( - 7 \beta + 25) q^{29} + ( - 18 \beta + 6) q^{31} + (3 \beta - 111) q^{33} + ( - 25 \beta + 17) q^{37} + (3 \beta - 147) q^{39} + ( - 6 \beta + 172) q^{41} + 108 q^{43} + (14 \beta + 438) q^{47} + (6 \beta + 819) q^{49} + (15 \beta + 117) q^{51} + ( - \beta + 317) q^{53} + ( - 24 \beta + 120) q^{57} + (49 \beta - 333) q^{59} + (32 \beta + 122) q^{61} + ( - 27 \beta - 9) q^{63} + ( - 34 \beta - 490) q^{67} + (24 \beta + 60) q^{69} + ( - 38 \beta - 154) q^{71} + (38 \beta + 706) q^{73} + (110 \beta - 350) q^{77} + ( - 14 \beta - 526) q^{79} + 81 q^{81} + (16 \beta + 124) q^{83} + ( - 21 \beta + 75) q^{87} + (48 \beta + 342) q^{89} + (146 \beta - 338) q^{91} + ( - 54 \beta + 18) q^{93} + ( - 32 \beta + 920) q^{97} + (9 \beta - 333) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} - 74 q^{11} - 98 q^{13} + 78 q^{17} + 80 q^{19} - 6 q^{21} + 40 q^{23} + 54 q^{27} + 50 q^{29} + 12 q^{31} - 222 q^{33} + 34 q^{37} - 294 q^{39} + 344 q^{41} + 216 q^{43} + 876 q^{47} + 1638 q^{49} + 234 q^{51} + 634 q^{53} + 240 q^{57} - 666 q^{59} + 244 q^{61} - 18 q^{63} - 980 q^{67} + 120 q^{69} - 308 q^{71} + 1412 q^{73} - 700 q^{77} - 1052 q^{79} + 162 q^{81} + 248 q^{83} + 150 q^{87} + 684 q^{89} - 676 q^{91} + 36 q^{93} + 1840 q^{97} - 666 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
6.17891
−5.17891
0 3.00000 0 0 0 −35.0735 0 9.00000 0
1.2 0 3.00000 0 0 0 33.0735 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.a.bq 2
4.b odd 2 1 600.4.a.t 2
5.b even 2 1 1200.4.a.bo 2
5.c odd 4 2 240.4.f.g 4
12.b even 2 1 1800.4.a.bn 2
15.e even 4 2 720.4.f.i 4
20.d odd 2 1 600.4.a.v 2
20.e even 4 2 120.4.f.d 4
40.i odd 4 2 960.4.f.o 4
40.k even 4 2 960.4.f.n 4
60.h even 2 1 1800.4.a.bl 2
60.l odd 4 2 360.4.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.f.d 4 20.e even 4 2
240.4.f.g 4 5.c odd 4 2
360.4.f.d 4 60.l odd 4 2
600.4.a.t 2 4.b odd 2 1
600.4.a.v 2 20.d odd 2 1
720.4.f.i 4 15.e even 4 2
960.4.f.n 4 40.k even 4 2
960.4.f.o 4 40.i odd 4 2
1200.4.a.bo 2 5.b even 2 1
1200.4.a.bq 2 1.a even 1 1 trivial
1800.4.a.bl 2 60.h even 2 1
1800.4.a.bn 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7}^{2} + 2T_{7} - 1160 \) Copy content Toggle raw display
\( T_{11}^{2} + 74T_{11} + 1240 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 1160 \) Copy content Toggle raw display
$11$ \( T^{2} + 74T + 1240 \) Copy content Toggle raw display
$13$ \( T^{2} + 98T + 2272 \) Copy content Toggle raw display
$17$ \( T^{2} - 78T - 1704 \) Copy content Toggle raw display
$19$ \( T^{2} - 80T - 6656 \) Copy content Toggle raw display
$23$ \( T^{2} - 40T - 7856 \) Copy content Toggle raw display
$29$ \( T^{2} - 50T - 5696 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T - 41760 \) Copy content Toggle raw display
$37$ \( T^{2} - 34T - 80336 \) Copy content Toggle raw display
$41$ \( T^{2} - 344T + 24940 \) Copy content Toggle raw display
$43$ \( (T - 108)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 876T + 166560 \) Copy content Toggle raw display
$53$ \( T^{2} - 634T + 100360 \) Copy content Toggle raw display
$59$ \( T^{2} + 666T - 198840 \) Copy content Toggle raw display
$61$ \( T^{2} - 244T - 117212 \) Copy content Toggle raw display
$67$ \( T^{2} + 980T + 90976 \) Copy content Toggle raw display
$71$ \( T^{2} + 308T - 162560 \) Copy content Toggle raw display
$73$ \( T^{2} - 1412 T + 312160 \) Copy content Toggle raw display
$79$ \( T^{2} + 1052 T + 251392 \) Copy content Toggle raw display
$83$ \( T^{2} - 248T - 17648 \) Copy content Toggle raw display
$89$ \( T^{2} - 684T - 180252 \) Copy content Toggle raw display
$97$ \( T^{2} - 1840 T + 714304 \) Copy content Toggle raw display
show more
show less