Properties

Label 600.4.a.u
Level $600$
Weight $4$
Character orbit 600.a
Self dual yes
Analytic conductor $35.401$
Analytic rank $1$
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] = [600,4,Mod(1,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, 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("600.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.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: \(35.4011460034\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(\beta = 2\sqrt{109}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - \beta - 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + ( - \beta - 1) q^{7} + 9 q^{9} + (3 \beta - 8) q^{11} + ( - 2 \beta - 41) q^{13} + (5 \beta - 4) q^{17} + (\beta - 105) q^{19} + ( - 3 \beta - 3) q^{21} + (\beta - 20) q^{23} + 27 q^{27} + (\beta - 120) q^{29} + ( - 9 \beta + 109) q^{31} + (9 \beta - 24) q^{33} + ( - 10 \beta - 182) q^{37} + ( - 6 \beta - 123) q^{39} + (3 \beta + 42) q^{41} + ( - 5 \beta - 137) q^{43} + ( - 12 \beta - 262) q^{47} + (2 \beta + 94) q^{49} + (15 \beta - 12) q^{51} + ( - 3 \beta - 222) q^{53} + (3 \beta - 315) q^{57} + ( - 8 \beta - 542) q^{59} + (4 \beta + 387) q^{61} + ( - 9 \beta - 9) q^{63} + (37 \beta - 105) q^{67} + (3 \beta - 60) q^{69} + ( - 29 \beta + 554) q^{71} + ( - 6 \beta - 246) q^{73} + (5 \beta - 1300) q^{77} + (28 \beta - 664) q^{79} + 81 q^{81} + (32 \beta + 14) q^{83} + (3 \beta - 360) q^{87} + ( - 4 \beta + 712) q^{89} + (43 \beta + 913) q^{91} + ( - 27 \beta + 327) q^{93} + ( - 16 \beta - 1185) q^{97} + (27 \beta - 72) q^{99}+O(q^{100}) \) 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 \( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} - 16 q^{11} - 82 q^{13} - 8 q^{17} - 210 q^{19} - 6 q^{21} - 40 q^{23} + 54 q^{27} - 240 q^{29} + 218 q^{31} - 48 q^{33} - 364 q^{37} - 246 q^{39} + 84 q^{41} - 274 q^{43} - 524 q^{47} + 188 q^{49} - 24 q^{51} - 444 q^{53} - 630 q^{57} - 1084 q^{59} + 774 q^{61} - 18 q^{63} - 210 q^{67} - 120 q^{69} + 1108 q^{71} - 492 q^{73} - 2600 q^{77} - 1328 q^{79} + 162 q^{81} + 28 q^{83} - 720 q^{87} + 1424 q^{89} + 1826 q^{91} + 654 q^{93} - 2370 q^{97} - 144 q^{99}+O(q^{100}) \) 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
5.72015
−4.72015
0 3.00000 0 0 0 −21.8806 0 9.00000 0
1.2 0 3.00000 0 0 0 19.8806 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 600.4.a.u yes 2
3.b odd 2 1 1800.4.a.bm 2
4.b odd 2 1 1200.4.a.bp 2
5.b even 2 1 600.4.a.s 2
5.c odd 4 2 600.4.f.j 4
15.d odd 2 1 1800.4.a.bo 2
15.e even 4 2 1800.4.f.z 4
20.d odd 2 1 1200.4.a.br 2
20.e even 4 2 1200.4.f.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.s 2 5.b even 2 1
600.4.a.u yes 2 1.a even 1 1 trivial
600.4.f.j 4 5.c odd 4 2
1200.4.a.bp 2 4.b odd 2 1
1200.4.a.br 2 20.d odd 2 1
1200.4.f.x 4 20.e even 4 2
1800.4.a.bm 2 3.b odd 2 1
1800.4.a.bo 2 15.d odd 2 1
1800.4.f.z 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(600))\):

\( T_{7}^{2} + 2T_{7} - 435 \) Copy content Toggle raw display
\( T_{11}^{2} + 16T_{11} - 3860 \) 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
$7$ \( T^{2} + 2T - 435 \) Copy content Toggle raw display
$11$ \( T^{2} + 16T - 3860 \) Copy content Toggle raw display
$13$ \( T^{2} + 82T - 63 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T - 10884 \) Copy content Toggle raw display
$19$ \( T^{2} + 210T + 10589 \) Copy content Toggle raw display
$23$ \( T^{2} + 40T - 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 240T + 13964 \) Copy content Toggle raw display
$31$ \( T^{2} - 218T - 23435 \) Copy content Toggle raw display
$37$ \( T^{2} + 364T - 10476 \) Copy content Toggle raw display
$41$ \( T^{2} - 84T - 2160 \) Copy content Toggle raw display
$43$ \( T^{2} + 274T + 7869 \) Copy content Toggle raw display
$47$ \( T^{2} + 524T + 5860 \) Copy content Toggle raw display
$53$ \( T^{2} + 444T + 45360 \) Copy content Toggle raw display
$59$ \( T^{2} + 1084 T + 265860 \) Copy content Toggle raw display
$61$ \( T^{2} - 774T + 142793 \) Copy content Toggle raw display
$67$ \( T^{2} + 210T - 585859 \) Copy content Toggle raw display
$71$ \( T^{2} - 1108T - 59760 \) Copy content Toggle raw display
$73$ \( T^{2} + 492T + 44820 \) Copy content Toggle raw display
$79$ \( T^{2} + 1328T + 99072 \) Copy content Toggle raw display
$83$ \( T^{2} - 28T - 446268 \) Copy content Toggle raw display
$89$ \( T^{2} - 1424 T + 499968 \) Copy content Toggle raw display
$97$ \( T^{2} + 2370 T + 1292609 \) Copy content Toggle raw display
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