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Properties

Label 600.3.p.a
Level $600$
Weight $3$
Character orbit 600.p
Analytic conductor $16.349$
Analytic rank $0$
Dimension $8$
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] = [600,3,Mod(499,600)]
 
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([1, 1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.499");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.p (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: \(16.3488158616\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{2}) q^{2} - \beta_{4} q^{3} + (\beta_{7} - \beta_{3} + 2) q^{4} + (\beta_{7} - 1) q^{6} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 2 \beta_1) q^{7}+ \cdots - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{2}) q^{2} - \beta_{4} q^{3} + (\beta_{7} - \beta_{3} + 2) q^{4} + (\beta_{7} - 1) q^{6} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 2 \beta_1) q^{7}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} - 12 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} - 12 q^{6} - 24 q^{9} - 64 q^{11} - 72 q^{14} - 16 q^{16} - 64 q^{19} - 72 q^{24} - 192 q^{26} - 88 q^{34} - 48 q^{36} + 80 q^{41} - 128 q^{44} - 144 q^{46} + 88 q^{49} - 192 q^{51} + 36 q^{54} - 336 q^{56} + 256 q^{59} + 64 q^{64} + 96 q^{66} + 240 q^{74} - 32 q^{76} + 72 q^{81} + 48 q^{84} + 176 q^{86} + 400 q^{89} + 576 q^{91} + 336 q^{94} - 48 q^{96} + 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + \nu^{5} - 4\nu^{4} + \nu^{3} + 6\nu^{2} - 10\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 5\nu^{3} - 4\nu^{2} + 14\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 13\nu^{3} + 13\nu^{2} - 38\nu + 26 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} - 11\nu^{6} + 6\nu^{5} + 17\nu^{4} - 24\nu^{3} - 15\nu^{2} + 62\nu - 48 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 3\nu^{7} - 8\nu^{6} + 4\nu^{5} + 12\nu^{4} - 17\nu^{3} - 13\nu^{2} + 46\nu - 33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 19\nu^{6} - 11\nu^{5} - 27\nu^{4} + 41\nu^{3} + 25\nu^{2} - 110\nu + 86 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 5\beta_{5} + 3\beta_{4} - \beta_{3} + 4\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} + \beta_{5} + 7\beta_{4} - 7\beta_{3} + 4\beta_{2} + 3\beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{5} + \beta_{4} + 8\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{7} - 8\beta_{6} - 7\beta_{5} - 13\beta_{4} - 9\beta_{3} + 2\beta_{2} + 5\beta _1 + 14 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
1.40994 − 0.109843i
1.40994 + 0.109843i
0.665665 − 1.24775i
0.665665 + 1.24775i
1.20036 + 0.747754i
1.20036 − 0.747754i
−1.27597 + 0.609843i
−1.27597 − 0.609843i
−1.96622 − 0.366025i − 1.73205i 3.73205 + 1.43937i 0 −0.633975 + 3.40559i 10.7436 −6.81119 − 4.19615i −3.00000 0
499.2 −1.96622 + 0.366025i 1.73205i 3.73205 − 1.43937i 0 −0.633975 − 3.40559i 10.7436 −6.81119 + 4.19615i −3.00000 0
499.3 −1.46081 − 1.36603i − 1.73205i 0.267949 + 3.99102i 0 −2.36603 + 2.53020i −2.13878 5.06040 − 6.19615i −3.00000 0
499.4 −1.46081 + 1.36603i 1.73205i 0.267949 − 3.99102i 0 −2.36603 − 2.53020i −2.13878 5.06040 + 6.19615i −3.00000 0
499.5 1.46081 − 1.36603i − 1.73205i 0.267949 − 3.99102i 0 −2.36603 − 2.53020i 2.13878 −5.06040 − 6.19615i −3.00000 0
499.6 1.46081 + 1.36603i 1.73205i 0.267949 + 3.99102i 0 −2.36603 + 2.53020i 2.13878 −5.06040 + 6.19615i −3.00000 0
499.7 1.96622 − 0.366025i − 1.73205i 3.73205 − 1.43937i 0 −0.633975 − 3.40559i −10.7436 6.81119 − 4.19615i −3.00000 0
499.8 1.96622 + 0.366025i 1.73205i 3.73205 + 1.43937i 0 −0.633975 + 3.40559i −10.7436 6.81119 + 4.19615i −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 499.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
40.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.3.p.a 8
4.b odd 2 1 2400.3.p.a 8
5.b even 2 1 inner 600.3.p.a 8
5.c odd 4 1 24.3.b.a ✓ 4
5.c odd 4 1 600.3.g.a 4
8.b even 2 1 2400.3.p.a 8
8.d odd 2 1 inner 600.3.p.a 8
15.e even 4 1 72.3.b.b 4
20.d odd 2 1 2400.3.p.a 8
20.e even 4 1 96.3.b.a 4
20.e even 4 1 2400.3.g.a 4
40.e odd 2 1 inner 600.3.p.a 8
40.f even 2 1 2400.3.p.a 8
40.i odd 4 1 96.3.b.a 4
40.i odd 4 1 2400.3.g.a 4
40.k even 4 1 24.3.b.a ✓ 4
40.k even 4 1 600.3.g.a 4
60.l odd 4 1 288.3.b.b 4
80.i odd 4 1 768.3.g.h 8
80.j even 4 1 768.3.g.h 8
80.s even 4 1 768.3.g.h 8
80.t odd 4 1 768.3.g.h 8
120.q odd 4 1 72.3.b.b 4
120.w even 4 1 288.3.b.b 4
240.z odd 4 1 2304.3.g.z 8
240.bb even 4 1 2304.3.g.z 8
240.bd odd 4 1 2304.3.g.z 8
240.bf even 4 1 2304.3.g.z 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a ✓ 4 5.c odd 4 1
24.3.b.a ✓ 4 40.k even 4 1
72.3.b.b 4 15.e even 4 1
72.3.b.b 4 120.q odd 4 1
96.3.b.a 4 20.e even 4 1
96.3.b.a 4 40.i odd 4 1
288.3.b.b 4 60.l odd 4 1
288.3.b.b 4 120.w even 4 1
600.3.g.a 4 5.c odd 4 1
600.3.g.a 4 40.k even 4 1
600.3.p.a 8 1.a even 1 1 trivial
600.3.p.a 8 5.b even 2 1 inner
600.3.p.a 8 8.d odd 2 1 inner
600.3.p.a 8 40.e odd 2 1 inner
768.3.g.h 8 80.i odd 4 1
768.3.g.h 8 80.j even 4 1
768.3.g.h 8 80.s even 4 1
768.3.g.h 8 80.t odd 4 1
2304.3.g.z 8 240.z odd 4 1
2304.3.g.z 8 240.bb even 4 1
2304.3.g.z 8 240.bd odd 4 1
2304.3.g.z 8 240.bf even 4 1
2400.3.g.a 4 20.e even 4 1
2400.3.g.a 4 40.i odd 4 1
2400.3.p.a 8 4.b odd 2 1
2400.3.p.a 8 8.b even 2 1
2400.3.p.a 8 20.d odd 2 1
2400.3.p.a 8 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 120T_{7}^{2} + 528 \) acting on \(S_{3}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 8 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 120 T^{2} + 528)^{2} \) Copy content Toggle raw display
$11$ \( (T + 8)^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 384 T^{2} + 33792)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 392 T^{2} + 35344)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16 T + 16)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 480 T^{2} + 8448)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1608 T^{2} + 528)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3384 T^{2} + 279312)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 864 T^{2} + 76032)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 20 T - 1628)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 992 T^{2} + 135424)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 3552 T^{2} + 8448)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 1800 T^{2} + 803088)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 64 T + 592)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 3552 T^{2} + 8448)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9056 T^{2} + 13424896)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8928 T^{2} + 12849408)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 11144 T^{2} + 327184)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 7416 T^{2} + 10797072)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4736 T^{2} + 692224)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 100 T - 4412)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 14216 T^{2} + 45104656)^{2} \) Copy content Toggle raw display
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