Properties

Label 600.2.w.f
Level $600$
Weight $2$
Character orbit 600.w
Analytic conductor $4.791$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,2,Mod(293,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.w (of order \(4\), degree \(2\), 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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1) q^{4} + (\beta_{2} + \beta_1 - 2) q^{6} + (3 \beta_{2} + 3) q^{7} + (2 \beta_{2} + 2) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1) q^{4} + (\beta_{2} + \beta_1 - 2) q^{6} + (3 \beta_{2} + 3) q^{7} + (2 \beta_{2} + 2) q^{8} - 3 q^{9} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{12} + (2 \beta_{2} - 4 \beta_1 + 2) q^{13} + ( - 3 \beta_{3} + 3 \beta_1) q^{14} + ( - 2 \beta_{3} + 2 \beta_1) q^{16} + ( - 4 \beta_{2} + 4) q^{17} + (3 \beta_{2} - 3 \beta_1) q^{18} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{19} + ( - 6 \beta_{3} + 3 \beta_{2} - 3) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} - 4) q^{22} + (\beta_{2} + 1) q^{23} + ( - 4 \beta_{3} + 2 \beta_{2} - 2) q^{24} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{26} + (3 \beta_{3} - 3 \beta_1 + 3) q^{27} + (6 \beta_1 - 6) q^{28} + (2 \beta_{3} - 2 \beta_1 + 2) q^{29} - 4 q^{31} + (4 \beta_1 - 4) q^{32} - 6 \beta_{2} q^{33} + (4 \beta_{3} - 8 \beta_{2} + 4 \beta_1) q^{34} + ( - 3 \beta_{3} - 3 \beta_1) q^{36} + (2 \beta_{3} + 2 \beta_{2} + 4) q^{38} + ( - 6 \beta_{2} + 6) q^{39} + 6 \beta_{2} q^{41} + ( - 3 \beta_{3} + 3 \beta_1 - 12) q^{42} + ( - \beta_{2} + 2 \beta_1 - 1) q^{43} + (2 \beta_{3} - 2 \beta_1 - 4) q^{44} + ( - \beta_{3} + \beta_1) q^{46} + ( - 5 \beta_{2} + 5) q^{47} + ( - 2 \beta_{3} + 2 \beta_1 - 8) q^{48} + 11 \beta_{2} q^{49} + ( - 4 \beta_{2} + 8 \beta_1 - 4) q^{51} + (4 \beta_{3} - 8 \beta_{2} - 4) q^{52} + ( - 4 \beta_{3} + 2 \beta_{2} - 2) q^{53} + ( - 3 \beta_{2} - 3 \beta_1 + 6) q^{54} + 12 \beta_{2} q^{56} + 6 \beta_{2} q^{57} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{58} + ( - 6 \beta_{3} + 6 \beta_1 - 6) q^{59} + (2 \beta_{3} - 2 \beta_1 + 2) q^{61} + (4 \beta_{2} - 4 \beta_1) q^{62} + ( - 9 \beta_{2} - 9) q^{63} + 8 \beta_{2} q^{64} + (6 \beta_{3} - 6 \beta_{2}) q^{66} + (6 \beta_{3} - 3 \beta_{2} + 3) q^{67} + (8 \beta_{3} + 8) q^{68} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{69} - 12 \beta_{2} q^{71} + ( - 6 \beta_{2} - 6) q^{72} + ( - 6 \beta_{2} + 6) q^{73} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{76} + (6 \beta_{2} - 12 \beta_1 + 6) q^{77} + (6 \beta_{3} - 12 \beta_{2} + 6 \beta_1) q^{78} - 8 \beta_{2} q^{79} + 9 q^{81} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{82} + (2 \beta_{3} - \beta_{2} + 1) q^{83} + (12 \beta_{2} - 6 \beta_1 - 6) q^{84} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{86} + 6 q^{87} + (4 \beta_{2} - 8 \beta_1 + 4) q^{88} - 12 q^{89} + (12 \beta_{3} - 12 \beta_1 + 12) q^{91} + (2 \beta_1 - 2) q^{92} + (4 \beta_{3} - 4 \beta_1 + 4) q^{93} + (5 \beta_{3} - 10 \beta_{2} + 5 \beta_1) q^{94} + (8 \beta_{2} - 4 \beta_1 - 4) q^{96} + (6 \beta_{2} + 6) q^{97} + ( - 11 \beta_{3} + 11 \beta_{2}) q^{98} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 6 q^{6} + 12 q^{7} + 8 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 6 q^{6} + 12 q^{7} + 8 q^{8} - 12 q^{9} + 12 q^{14} + 8 q^{16} + 16 q^{17} - 6 q^{18} - 12 q^{22} + 4 q^{23} - 12 q^{28} - 16 q^{31} - 8 q^{32} + 12 q^{38} + 24 q^{39} - 36 q^{42} - 24 q^{44} + 4 q^{46} + 20 q^{47} - 24 q^{48} - 24 q^{52} + 18 q^{54} + 12 q^{58} - 8 q^{62} - 36 q^{63} - 12 q^{66} + 16 q^{68} - 24 q^{72} + 24 q^{73} + 24 q^{76} + 36 q^{81} + 12 q^{82} - 36 q^{84} + 24 q^{87} - 48 q^{89} - 4 q^{92} - 24 q^{96} + 24 q^{97} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.366025 + 1.36603i 1.73205i −1.73205 1.00000i 0 −2.36603 0.633975i 3.00000 3.00000i 2.00000 2.00000i −3.00000 0
293.2 1.36603 0.366025i 1.73205i 1.73205 1.00000i 0 −0.633975 2.36603i 3.00000 3.00000i 2.00000 2.00000i −3.00000 0
557.1 −0.366025 1.36603i 1.73205i −1.73205 + 1.00000i 0 −2.36603 + 0.633975i 3.00000 + 3.00000i 2.00000 + 2.00000i −3.00000 0
557.2 1.36603 + 0.366025i 1.73205i 1.73205 + 1.00000i 0 −0.633975 + 2.36603i 3.00000 + 3.00000i 2.00000 + 2.00000i −3.00000 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
8.b even 2 1 inner
15.e even 4 1 inner
120.w even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.w.f yes 4
3.b odd 2 1 600.2.w.d yes 4
5.b even 2 1 600.2.w.c 4
5.c odd 4 1 600.2.w.d yes 4
5.c odd 4 1 600.2.w.e yes 4
8.b even 2 1 inner 600.2.w.f yes 4
15.d odd 2 1 600.2.w.e yes 4
15.e even 4 1 600.2.w.c 4
15.e even 4 1 inner 600.2.w.f yes 4
24.h odd 2 1 600.2.w.d yes 4
40.f even 2 1 600.2.w.c 4
40.i odd 4 1 600.2.w.d yes 4
40.i odd 4 1 600.2.w.e yes 4
120.i odd 2 1 600.2.w.e yes 4
120.w even 4 1 600.2.w.c 4
120.w even 4 1 inner 600.2.w.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.w.c 4 5.b even 2 1
600.2.w.c 4 15.e even 4 1
600.2.w.c 4 40.f even 2 1
600.2.w.c 4 120.w even 4 1
600.2.w.d yes 4 3.b odd 2 1
600.2.w.d yes 4 5.c odd 4 1
600.2.w.d yes 4 24.h odd 2 1
600.2.w.d yes 4 40.i odd 4 1
600.2.w.e yes 4 5.c odd 4 1
600.2.w.e yes 4 15.d odd 2 1
600.2.w.e yes 4 40.i odd 4 1
600.2.w.e yes 4 120.i odd 2 1
600.2.w.f yes 4 1.a even 1 1 trivial
600.2.w.f yes 4 8.b even 2 1 inner
600.2.w.f yes 4 15.e even 4 1 inner
600.2.w.f yes 4 120.w even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{2} - 6T_{7} + 18 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{17}^{2} - 8T_{17} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 576 \) Copy content Toggle raw display
$17$ \( (T^{2} - 8 T + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 36 \) Copy content Toggle raw display
$47$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 576 \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 2916 \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T + 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 36 \) Copy content Toggle raw display
$89$ \( (T + 12)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T + 72)^{2} \) Copy content Toggle raw display
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