Properties

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

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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: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 15)
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 \(\beta = \sqrt{41}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - 3 \beta - 3) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - 3 \beta - 3) q^{7} + 9 q^{9} + ( - 3 \beta + 21) q^{11} + (3 \beta + 39) q^{13} + ( - 5 \beta + 51) q^{17} + ( - 12 \beta - 28) q^{19} + (9 \beta + 9) q^{21} + (4 \beta + 24) q^{23} - 27 q^{27} + ( - 21 \beta - 159) q^{29} + ( - 6 \beta - 26) q^{31} + (9 \beta - 63) q^{33} + ( - 27 \beta + 153) q^{37} + ( - 9 \beta - 117) q^{39} + (6 \beta - 204) q^{41} + (48 \beta - 60) q^{43} + ( - 46 \beta - 90) q^{47} + (18 \beta + 35) q^{49} + (15 \beta - 153) q^{51} + (41 \beta + 201) q^{53} + (36 \beta + 84) q^{57} + ( - 3 \beta + 93) q^{59} + ( - 48 \beta + 170) q^{61} + ( - 27 \beta - 27) q^{63} + (30 \beta - 366) q^{67} + ( - 12 \beta - 72) q^{69} + (90 \beta + 18) q^{71} + ( - 54 \beta + 666) q^{73} + ( - 54 \beta + 306) q^{77} + (150 \beta - 190) q^{79} + 81 q^{81} + ( - 104 \beta + 492) q^{83} + (63 \beta + 477) q^{87} + (72 \beta + 558) q^{89} + ( - 126 \beta - 486) q^{91} + (18 \beta + 78) q^{93} + (120 \beta - 384) q^{97} + ( - 27 \beta + 189) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9} + 42 q^{11} + 78 q^{13} + 102 q^{17} - 56 q^{19} + 18 q^{21} + 48 q^{23} - 54 q^{27} - 318 q^{29} - 52 q^{31} - 126 q^{33} + 306 q^{37} - 234 q^{39} - 408 q^{41} - 120 q^{43} - 180 q^{47} + 70 q^{49} - 306 q^{51} + 402 q^{53} + 168 q^{57} + 186 q^{59} + 340 q^{61} - 54 q^{63} - 732 q^{67} - 144 q^{69} + 36 q^{71} + 1332 q^{73} + 612 q^{77} - 380 q^{79} + 162 q^{81} + 984 q^{83} + 954 q^{87} + 1116 q^{89} - 972 q^{91} + 156 q^{93} - 768 q^{97} + 378 q^{99}+O(q^{100}) \) 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
3.70156
−2.70156
0 −3.00000 0 0 0 −22.2094 0 9.00000 0
1.2 0 −3.00000 0 0 0 16.2094 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.bn 2
4.b odd 2 1 75.4.a.f 2
5.b even 2 1 1200.4.a.bt 2
5.c odd 4 2 240.4.f.f 4
12.b even 2 1 225.4.a.i 2
15.e even 4 2 720.4.f.j 4
20.d odd 2 1 75.4.a.c 2
20.e even 4 2 15.4.b.a 4
40.i odd 4 2 960.4.f.p 4
40.k even 4 2 960.4.f.q 4
60.h even 2 1 225.4.a.o 2
60.l odd 4 2 45.4.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 20.e even 4 2
45.4.b.b 4 60.l odd 4 2
75.4.a.c 2 20.d odd 2 1
75.4.a.f 2 4.b odd 2 1
225.4.a.i 2 12.b even 2 1
225.4.a.o 2 60.h even 2 1
240.4.f.f 4 5.c odd 4 2
720.4.f.j 4 15.e even 4 2
960.4.f.p 4 40.i odd 4 2
960.4.f.q 4 40.k even 4 2
1200.4.a.bn 2 1.a even 1 1 trivial
1200.4.a.bt 2 5.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7}^{2} + 6T_{7} - 360 \) Copy content Toggle raw display
\( T_{11}^{2} - 42T_{11} + 72 \) 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
$7$ \( T^{2} + 6T - 360 \) Copy content Toggle raw display
$11$ \( T^{2} - 42T + 72 \) Copy content Toggle raw display
$13$ \( T^{2} - 78T + 1152 \) Copy content Toggle raw display
$17$ \( T^{2} - 102T + 1576 \) Copy content Toggle raw display
$19$ \( T^{2} + 56T - 5120 \) Copy content Toggle raw display
$23$ \( T^{2} - 48T - 80 \) Copy content Toggle raw display
$29$ \( T^{2} + 318T + 7200 \) Copy content Toggle raw display
$31$ \( T^{2} + 52T - 800 \) Copy content Toggle raw display
$37$ \( T^{2} - 306T - 6480 \) Copy content Toggle raw display
$41$ \( T^{2} + 408T + 40140 \) Copy content Toggle raw display
$43$ \( T^{2} + 120T - 90864 \) Copy content Toggle raw display
$47$ \( T^{2} + 180T - 78656 \) Copy content Toggle raw display
$53$ \( T^{2} - 402T - 28520 \) Copy content Toggle raw display
$59$ \( T^{2} - 186T + 8280 \) Copy content Toggle raw display
$61$ \( T^{2} - 340T - 65564 \) Copy content Toggle raw display
$67$ \( T^{2} + 732T + 97056 \) Copy content Toggle raw display
$71$ \( T^{2} - 36T - 331776 \) Copy content Toggle raw display
$73$ \( T^{2} - 1332 T + 324000 \) Copy content Toggle raw display
$79$ \( T^{2} + 380T - 886400 \) Copy content Toggle raw display
$83$ \( T^{2} - 984T - 201392 \) Copy content Toggle raw display
$89$ \( T^{2} - 1116T + 98820 \) Copy content Toggle raw display
$97$ \( T^{2} + 768T - 442944 \) Copy content Toggle raw display
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