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Properties

Label 1600.3.h.n
Level $1600$
Weight $3$
Character orbit 1600.h
Analytic conductor $43.597$
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] = [1600,3,Mod(1599,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1600.1599");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.h (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: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{6} + \beta_{3}) q^{7} + (\beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{6} + \beta_{3}) q^{7} + (\beta_{4} + 1) q^{9} + (\beta_{7} - \beta_{2}) q^{11} + ( - \beta_{5} + 2 \beta_1) q^{13} + (4 \beta_{5} + 3 \beta_1) q^{17} + \beta_{7} q^{19} + ( - 5 \beta_{4} - 10) q^{21} + ( - \beta_{6} - 3 \beta_{3}) q^{23} + ( - 2 \beta_{6} + 6 \beta_{3}) q^{27} + (2 \beta_{4} - 2) q^{29} + ( - \beta_{7} - 5 \beta_{2}) q^{31} + ( - 6 \beta_{5} + 10 \beta_1) q^{33} + (5 \beta_{5} + 2 \beta_1) q^{37} + (\beta_{7} - \beta_{2}) q^{39} + ( - 3 \beta_{4} - 28) q^{41} + ( - 2 \beta_{6} + 3 \beta_{3}) q^{43} + ( - 3 \beta_{6} + 13 \beta_{3}) q^{47} + (5 \beta_{4} + 1) q^{49} + ( - 4 \beta_{7} - 7 \beta_{2}) q^{51} + ( - 5 \beta_{5} + 22 \beta_1) q^{53} - 8 \beta_{5} q^{57} + ( - 3 \beta_{7} - 6 \beta_{2}) q^{59} + (13 \beta_{4} - 32) q^{61} + (\beta_{6} + 11 \beta_{3}) q^{63} + (10 \beta_{6} + 11 \beta_{3}) q^{67} + (7 \beta_{4} + 30) q^{69} + (5 \beta_{7} - \beta_{2}) q^{71} + (16 \beta_{5} + 33 \beta_1) q^{73} + ( - 10 \beta_{5} + 30 \beta_1) q^{77} + ( - 4 \beta_{7} - 10 \beta_{2}) q^{79} + ( - 7 \beta_{4} - 69) q^{81} + (6 \beta_{6} - 13 \beta_{3}) q^{83} + ( - 4 \beta_{6} - 2 \beta_{3}) q^{87} + ( - 20 \beta_{4} + 22) q^{89} + (2 \beta_{7} - 3 \beta_{2}) q^{91} + (18 \beta_{5} + 50 \beta_1) q^{93} + ( - 6 \beta_{5} + 33 \beta_1) q^{97} + ( - 3 \beta_{7} + 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 80 q^{21} - 16 q^{29} - 224 q^{41} + 8 q^{49} - 256 q^{61} + 240 q^{69} - 552 q^{81} + 176 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{20}^{5} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{20}^{6} + 4\zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{20}^{7} + 2\zeta_{20}^{5} - 2\zeta_{20}^{3} + 4\zeta_{20} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -4\zeta_{20}^{6} + 4\zeta_{20}^{4} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 4\zeta_{20}^{7} - 2\zeta_{20}^{5} + 4\zeta_{20}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -4\zeta_{20}^{7} + 4\zeta_{20}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 8\zeta_{20}^{6} - 8\zeta_{20}^{4} + 16\zeta_{20}^{2} - 8 \) Copy content Toggle raw display

Display \(\zeta_{20}^j\) in terms of \(\beta_i\)

\(\zeta_{20}\)\(=\) \( ( \beta_{5} + 2\beta_{3} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{4} + 4 ) / 16 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( ( \beta_{4} + \beta_{2} - 2 ) / 8 \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( ( -\beta_{4} + \beta_{2} + 2 ) / 8 \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_1 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{20}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1599.1
0.951057 − 0.309017i
0.951057 + 0.309017i
0.587785 − 0.809017i
0.587785 + 0.809017i
−0.587785 + 0.809017i
−0.587785 − 0.809017i
−0.951057 + 0.309017i
−0.951057 − 0.309017i
0 −3.80423 0 0 0 8.50651 0 5.47214 0
1599.2 0 −3.80423 0 0 0 8.50651 0 5.47214 0
1599.3 0 −2.35114 0 0 0 −5.25731 0 −3.47214 0
1599.4 0 −2.35114 0 0 0 −5.25731 0 −3.47214 0
1599.5 0 2.35114 0 0 0 5.25731 0 −3.47214 0
1599.6 0 2.35114 0 0 0 5.25731 0 −3.47214 0
1599.7 0 3.80423 0 0 0 −8.50651 0 5.47214 0
1599.8 0 3.80423 0 0 0 −8.50651 0 5.47214 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1599.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.3.h.n 8
4.b odd 2 1 inner 1600.3.h.n 8
5.b even 2 1 inner 1600.3.h.n 8
5.c odd 4 1 320.3.b.c 4
5.c odd 4 1 1600.3.b.s 4
8.b even 2 1 100.3.d.b 8
8.d odd 2 1 100.3.d.b 8
15.e even 4 1 2880.3.e.e 4
20.d odd 2 1 inner 1600.3.h.n 8
20.e even 4 1 320.3.b.c 4
20.e even 4 1 1600.3.b.s 4
24.f even 2 1 900.3.f.e 8
24.h odd 2 1 900.3.f.e 8
40.e odd 2 1 100.3.d.b 8
40.f even 2 1 100.3.d.b 8
40.i odd 4 1 20.3.b.a ✓ 4
40.i odd 4 1 100.3.b.f 4
40.k even 4 1 20.3.b.a ✓ 4
40.k even 4 1 100.3.b.f 4
60.l odd 4 1 2880.3.e.e 4
80.i odd 4 1 1280.3.g.e 8
80.j even 4 1 1280.3.g.e 8
80.s even 4 1 1280.3.g.e 8
80.t odd 4 1 1280.3.g.e 8
120.i odd 2 1 900.3.f.e 8
120.m even 2 1 900.3.f.e 8
120.q odd 4 1 180.3.c.a 4
120.q odd 4 1 900.3.c.k 4
120.w even 4 1 180.3.c.a 4
120.w even 4 1 900.3.c.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a ✓ 4 40.i odd 4 1
20.3.b.a ✓ 4 40.k even 4 1
100.3.b.f 4 40.i odd 4 1
100.3.b.f 4 40.k even 4 1
100.3.d.b 8 8.b even 2 1
100.3.d.b 8 8.d odd 2 1
100.3.d.b 8 40.e odd 2 1
100.3.d.b 8 40.f even 2 1
180.3.c.a 4 120.q odd 4 1
180.3.c.a 4 120.w even 4 1
320.3.b.c 4 5.c odd 4 1
320.3.b.c 4 20.e even 4 1
900.3.c.k 4 120.q odd 4 1
900.3.c.k 4 120.w even 4 1
900.3.f.e 8 24.f even 2 1
900.3.f.e 8 24.h odd 2 1
900.3.f.e 8 120.i odd 2 1
900.3.f.e 8 120.m even 2 1
1280.3.g.e 8 80.i odd 4 1
1280.3.g.e 8 80.j even 4 1
1280.3.g.e 8 80.s even 4 1
1280.3.g.e 8 80.t odd 4 1
1600.3.b.s 4 5.c odd 4 1
1600.3.b.s 4 20.e even 4 1
1600.3.h.n 8 1.a even 1 1 trivial
1600.3.h.n 8 4.b odd 2 1 inner
1600.3.h.n 8 5.b even 2 1 inner
1600.3.h.n 8 20.d odd 2 1 inner
2880.3.e.e 4 15.e even 4 1
2880.3.e.e 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} - 20T_{3}^{2} + 80 \) Copy content Toggle raw display
\( T_{7}^{4} - 100T_{7}^{2} + 2000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 20 T^{2} + 80)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 100 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 400 T^{2} + 1280)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 72 T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 712 T^{2} + 80656)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 320 T^{2} + 20480)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 260 T^{2} + 80)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 76)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2320 T^{2} + 154880)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1032 T^{2} + 234256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 56 T + 604)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 500 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4100 T^{2} + 3561680)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 4872 T^{2} + 2062096)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 5760 T^{2} + 1658880)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 64 T - 2356)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 10420 T^{2} + 19920080)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8080 T^{2} + 10138880)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 18952 T^{2} + 583696)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 13120 T^{2} + 2478080)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 6260 T^{2} + 2620880)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 44 T - 7516)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 10152 T^{2} + 13220496)^{2} \) Copy content Toggle raw display
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