Properties

Label 600.6.f.h
Level $600$
Weight $6$
Character orbit 600.f
Analytic conductor $96.230$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,6,Mod(49,600)]
 
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, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 600.f (of order \(2\), degree \(1\), not 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: \(96.2302918878\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 i q^{3} + 120 i q^{7} - 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 9 i q^{3} + 120 i q^{7} - 81 q^{9} + 524 q^{11} + 962 i q^{13} - 1358 i q^{17} + 2284 q^{19} + 1080 q^{21} - 2552 i q^{23} + 729 i q^{27} - 3966 q^{29} - 2992 q^{31} - 4716 i q^{33} + 13206 i q^{37} + 8658 q^{39} - 15126 q^{41} + 7316 i q^{43} - 6960 i q^{47} + 2407 q^{49} - 12222 q^{51} + 17482 i q^{53} - 20556 i q^{57} - 33884 q^{59} + 39118 q^{61} - 9720 i q^{63} + 32996 i q^{67} - 22968 q^{69} + 14248 q^{71} + 35990 i q^{73} + 62880 i q^{77} + 29888 q^{79} + 6561 q^{81} + 51884 i q^{83} + 35694 i q^{87} - 30714 q^{89} - 115440 q^{91} + 26928 i q^{93} - 48478 i q^{97} - 42444 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 162 q^{9} + 1048 q^{11} + 4568 q^{19} + 2160 q^{21} - 7932 q^{29} - 5984 q^{31} + 17316 q^{39} - 30252 q^{41} + 4814 q^{49} - 24444 q^{51} - 67768 q^{59} + 78236 q^{61} - 45936 q^{69} + 28496 q^{71} + 59776 q^{79} + 13122 q^{81} - 61428 q^{89} - 230880 q^{91} - 84888 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 9.00000i 0 0 0 120.000i 0 −81.0000 0
49.2 0 9.00000i 0 0 0 120.000i 0 −81.0000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.6.f.h 2
5.b even 2 1 inner 600.6.f.h 2
5.c odd 4 1 24.6.a.c 1
5.c odd 4 1 600.6.a.a 1
15.e even 4 1 72.6.a.b 1
20.e even 4 1 48.6.a.b 1
40.i odd 4 1 192.6.a.b 1
40.k even 4 1 192.6.a.j 1
60.l odd 4 1 144.6.a.d 1
80.i odd 4 1 768.6.d.f 2
80.j even 4 1 768.6.d.m 2
80.s even 4 1 768.6.d.m 2
80.t odd 4 1 768.6.d.f 2
120.q odd 4 1 576.6.a.ba 1
120.w even 4 1 576.6.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.6.a.c 1 5.c odd 4 1
48.6.a.b 1 20.e even 4 1
72.6.a.b 1 15.e even 4 1
144.6.a.d 1 60.l odd 4 1
192.6.a.b 1 40.i odd 4 1
192.6.a.j 1 40.k even 4 1
576.6.a.ba 1 120.q odd 4 1
576.6.a.bb 1 120.w even 4 1
600.6.a.a 1 5.c odd 4 1
600.6.f.h 2 1.a even 1 1 trivial
600.6.f.h 2 5.b even 2 1 inner
768.6.d.f 2 80.i odd 4 1
768.6.d.f 2 80.t odd 4 1
768.6.d.m 2 80.j even 4 1
768.6.d.m 2 80.s even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 14400 \) acting on \(S_{6}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 14400 \) Copy content Toggle raw display
$11$ \( (T - 524)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 925444 \) Copy content Toggle raw display
$17$ \( T^{2} + 1844164 \) Copy content Toggle raw display
$19$ \( (T - 2284)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6512704 \) Copy content Toggle raw display
$29$ \( (T + 3966)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2992)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 174398436 \) Copy content Toggle raw display
$41$ \( (T + 15126)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 53523856 \) Copy content Toggle raw display
$47$ \( T^{2} + 48441600 \) Copy content Toggle raw display
$53$ \( T^{2} + 305620324 \) Copy content Toggle raw display
$59$ \( (T + 33884)^{2} \) Copy content Toggle raw display
$61$ \( (T - 39118)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1088736016 \) Copy content Toggle raw display
$71$ \( (T - 14248)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1295280100 \) Copy content Toggle raw display
$79$ \( (T - 29888)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2691949456 \) Copy content Toggle raw display
$89$ \( (T + 30714)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2350116484 \) Copy content Toggle raw display
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