Properties

Label 1200.4.a.bl
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: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 75)
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 \(\beta = 4\sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + (\beta - 13) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + (\beta - 13) q^{7} + 9 q^{9} + (\beta - 14) q^{11} + (4 \beta + 9) q^{13} + (5 \beta - 34) q^{17} + ( - \beta - 3) q^{19} + ( - 3 \beta + 39) q^{21} + ( - 3 \beta - 66) q^{23} - 27 q^{27} + (7 \beta + 46) q^{29} + (7 \beta - 61) q^{31} + ( - 3 \beta + 42) q^{33} + ( - 6 \beta - 142) q^{37} + ( - 12 \beta - 27) q^{39} + (13 \beta + 196) q^{41} + ( - \beta - 345) q^{43} + ( - 8 \beta + 310) q^{47} + ( - 26 \beta + 130) q^{49} + ( - 15 \beta + 102) q^{51} + ( - 7 \beta - 424) q^{53} + (3 \beta + 9) q^{57} + (16 \beta - 62) q^{59} + ( - 14 \beta + 375) q^{61} + (9 \beta - 117) q^{63} + (25 \beta + 179) q^{67} + (9 \beta + 198) q^{69} + (5 \beta - 412) q^{71} + ( - 2 \beta - 54) q^{73} + ( - 27 \beta + 486) q^{77} + (40 \beta + 440) q^{79} + 81 q^{81} + (48 \beta - 78) q^{83} + ( - 21 \beta - 138) q^{87} + (36 \beta - 432) q^{89} + ( - 43 \beta + 1099) q^{91} + ( - 21 \beta + 183) q^{93} + 521 q^{97} + (9 \beta - 126) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 26 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 26 q^{7} + 18 q^{9} - 28 q^{11} + 18 q^{13} - 68 q^{17} - 6 q^{19} + 78 q^{21} - 132 q^{23} - 54 q^{27} + 92 q^{29} - 122 q^{31} + 84 q^{33} - 284 q^{37} - 54 q^{39} + 392 q^{41} - 690 q^{43} + 620 q^{47} + 260 q^{49} + 204 q^{51} - 848 q^{53} + 18 q^{57} - 124 q^{59} + 750 q^{61} - 234 q^{63} + 358 q^{67} + 396 q^{69} - 824 q^{71} - 108 q^{73} + 972 q^{77} + 880 q^{79} + 162 q^{81} - 156 q^{83} - 276 q^{87} - 864 q^{89} + 2198 q^{91} + 366 q^{93} + 1042 q^{97} - 252 q^{99}+O(q^{100}) \) 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
−4.35890
4.35890
0 −3.00000 0 0 0 −30.4356 0 9.00000 0
1.2 0 −3.00000 0 0 0 4.43560 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.bl 2
4.b odd 2 1 75.4.a.d 2
5.b even 2 1 1200.4.a.bu 2
5.c odd 4 2 1200.4.f.v 4
12.b even 2 1 225.4.a.n 2
20.d odd 2 1 75.4.a.e yes 2
20.e even 4 2 75.4.b.c 4
60.h even 2 1 225.4.a.j 2
60.l odd 4 2 225.4.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 4.b odd 2 1
75.4.a.e yes 2 20.d odd 2 1
75.4.b.c 4 20.e even 4 2
225.4.a.j 2 60.h even 2 1
225.4.a.n 2 12.b even 2 1
225.4.b.h 4 60.l odd 4 2
1200.4.a.bl 2 1.a even 1 1 trivial
1200.4.a.bu 2 5.b even 2 1
1200.4.f.v 4 5.c odd 4 2

Hecke kernels

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

\( T_{7}^{2} + 26T_{7} - 135 \) Copy content Toggle raw display
\( T_{11}^{2} + 28T_{11} - 108 \) 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
$7$ \( T^{2} + 26T - 135 \) Copy content Toggle raw display
$11$ \( T^{2} + 28T - 108 \) Copy content Toggle raw display
$13$ \( T^{2} - 18T - 4783 \) Copy content Toggle raw display
$17$ \( T^{2} + 68T - 6444 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 295 \) Copy content Toggle raw display
$23$ \( T^{2} + 132T + 1620 \) Copy content Toggle raw display
$29$ \( T^{2} - 92T - 12780 \) Copy content Toggle raw display
$31$ \( T^{2} + 122T - 11175 \) Copy content Toggle raw display
$37$ \( T^{2} + 284T + 9220 \) Copy content Toggle raw display
$41$ \( T^{2} - 392T - 12960 \) Copy content Toggle raw display
$43$ \( T^{2} + 690T + 118721 \) Copy content Toggle raw display
$47$ \( T^{2} - 620T + 76644 \) Copy content Toggle raw display
$53$ \( T^{2} + 848T + 164880 \) Copy content Toggle raw display
$59$ \( T^{2} + 124T - 73980 \) Copy content Toggle raw display
$61$ \( T^{2} - 750T + 81041 \) Copy content Toggle raw display
$67$ \( T^{2} - 358T - 157959 \) Copy content Toggle raw display
$71$ \( T^{2} + 824T + 162144 \) Copy content Toggle raw display
$73$ \( T^{2} + 108T + 1700 \) Copy content Toggle raw display
$79$ \( T^{2} - 880T - 292800 \) Copy content Toggle raw display
$83$ \( T^{2} + 156T - 694332 \) Copy content Toggle raw display
$89$ \( T^{2} + 864T - 207360 \) Copy content Toggle raw display
$97$ \( (T - 521)^{2} \) Copy content Toggle raw display
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