↑

Properties

Label 1600.3.h.o
Level $1600$
Weight $3$
Character orbit 1600.h
Analytic conductor $43.597$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: 8.0.18084870400.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 40x^{4} + 17x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 100)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{7} + \beta_{3}) q^{7} + (\beta_{2} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{7} + \beta_{3}) q^{7} + (\beta_{2} + 4) q^{9} + \beta_{4} q^{11} - \beta_{6} q^{13} + (\beta_{6} - \beta_{5}) q^{17} + ( - 2 \beta_{4} + \beta_1) q^{19} + (\beta_{2} + 8) q^{21} + (2 \beta_{7} + 4 \beta_{3}) q^{23} + (\beta_{7} + 8 \beta_{3}) q^{27} + ( - \beta_{2} + 10) q^{29} + 2 \beta_{4} q^{31} + \beta_{5} q^{33} + ( - \beta_{6} - 6 \beta_{5}) q^{37} + (\beta_{4} + 3 \beta_1) q^{39} + 17 q^{41} + (\beta_{7} + 11 \beta_{3}) q^{43} + ( - 3 \beta_{7} + 11 \beta_{3}) q^{47} + ( - 4 \beta_{2} + 19) q^{49} + ( - \beta_{4} - 2 \beta_1) q^{51} + (4 \beta_{6} + 6 \beta_{5}) q^{53} + ( - 5 \beta_{6} - 13 \beta_{5}) q^{57} + (9 \beta_{4} + 3 \beta_1) q^{59} + ( - 5 \beta_{2} + 28) q^{61} + ( - 8 \beta_{7} + 12 \beta_{3}) q^{63} + ( - 5 \beta_{7} - 6 \beta_{3}) q^{67} + (4 \beta_{2} + 42) q^{69} + ( - 7 \beta_{4} - 5 \beta_1) q^{71} + (\beta_{6} + 11 \beta_{5}) q^{73} + (5 \beta_{6} - 14 \beta_{5}) q^{77} + ( - 7 \beta_{4} + \beta_1) q^{79} + ( - \beta_{2} + 63) q^{81} + ( - 3 \beta_{7} - 14 \beta_{3}) q^{83} + ( - \beta_{7} - 3 \beta_{3}) q^{87} + (\beta_{2} + 55) q^{89} + ( - 10 \beta_{4} + 2 \beta_1) q^{91} + 2 \beta_{5} q^{93} + (6 \beta_{6} - 18 \beta_{5}) q^{97} + ( - 9 \beta_{4} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{9} + 64 q^{21} + 80 q^{29} + 136 q^{41} + 152 q^{49} + 224 q^{61} + 336 q^{69} + 504 q^{81} + 440 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} + 40x^{4} + 17x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{6} - 34\nu^{4} + 96\nu^{2} + 19 ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{6} + 68\nu^{4} + 28\nu^{2} - 698 ) / 55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{7} - 104\nu^{5} + 326\nu^{3} + 29\nu ) / 110 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -12\nu^{6} + 146\nu^{4} - 508\nu^{2} - 101 ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{7} - 104\nu^{5} + 326\nu^{3} + 249\nu ) / 22 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -46\nu^{7} + 556\nu^{5} - 1764\nu^{3} - 1346\nu ) / 55 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -39\nu^{7} + 480\nu^{5} - 1706\nu^{3} - 155\nu ) / 22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 5\beta_{3} ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{2} + 2\beta _1 + 60 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} + 15\beta_{6} + 32\beta_{5} - 115\beta_{3} ) / 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{4} + 5\beta_{2} + 5\beta _1 + 64 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -10\beta_{7} + 75\beta_{6} + 162\beta_{5} - 260\beta_{3} ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 85\beta_{4} + 185\beta_{2} + 439\beta _1 + 2370 ) / 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -25\beta_{7} + 595\beta_{6} + 1286\beta_{5} - 645\beta_{3} ) / 20 \) Copy content Toggle raw display

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

Character values

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

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

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
2.53999 + 0.500000i
2.53999 − 0.500000i
0.220086 + 0.500000i
0.220086 − 0.500000i
−0.220086 − 0.500000i
−0.220086 + 0.500000i
−2.53999 − 0.500000i
−2.53999 + 0.500000i
0 −5.07999 0 0 0 −4.09573 0 16.8062 0
1599.2 0 −5.07999 0 0 0 −4.09573 0 16.8062 0
1599.3 0 −0.440172 0 0 0 10.9190 0 −8.80625 0
1599.4 0 −0.440172 0 0 0 10.9190 0 −8.80625 0
1599.5 0 0.440172 0 0 0 −10.9190 0 −8.80625 0
1599.6 0 0.440172 0 0 0 −10.9190 0 −8.80625 0
1599.7 0 5.07999 0 0 0 4.09573 0 16.8062 0
1599.8 0 5.07999 0 0 0 4.09573 0 16.8062 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.o 8
4.b odd 2 1 inner 1600.3.h.o 8
5.b even 2 1 inner 1600.3.h.o 8
5.c odd 4 1 1600.3.b.o 4
5.c odd 4 1 1600.3.b.p 4
8.b even 2 1 100.3.d.a 8
8.d odd 2 1 100.3.d.a 8
20.d odd 2 1 inner 1600.3.h.o 8
20.e even 4 1 1600.3.b.o 4
20.e even 4 1 1600.3.b.p 4
24.f even 2 1 900.3.f.d 8
24.h odd 2 1 900.3.f.d 8
40.e odd 2 1 100.3.d.a 8
40.f even 2 1 100.3.d.a 8
40.i odd 4 1 100.3.b.d ✓ 4
40.i odd 4 1 100.3.b.e yes 4
40.k even 4 1 100.3.b.d ✓ 4
40.k even 4 1 100.3.b.e yes 4
120.i odd 2 1 900.3.f.d 8
120.m even 2 1 900.3.f.d 8
120.q odd 4 1 900.3.c.l 4
120.q odd 4 1 900.3.c.m 4
120.w even 4 1 900.3.c.l 4
120.w even 4 1 900.3.c.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.3.b.d ✓ 4 40.i odd 4 1
100.3.b.d ✓ 4 40.k even 4 1
100.3.b.e yes 4 40.i odd 4 1
100.3.b.e yes 4 40.k even 4 1
100.3.d.a 8 8.b even 2 1
100.3.d.a 8 8.d odd 2 1
100.3.d.a 8 40.e odd 2 1
100.3.d.a 8 40.f even 2 1
900.3.c.l 4 120.q odd 4 1
900.3.c.l 4 120.w even 4 1
900.3.c.m 4 120.q odd 4 1
900.3.c.m 4 120.w even 4 1
900.3.f.d 8 24.f even 2 1
900.3.f.d 8 24.h odd 2 1
900.3.f.d 8 120.i odd 2 1
900.3.f.d 8 120.m even 2 1
1600.3.b.o 4 5.c odd 4 1
1600.3.b.o 4 20.e even 4 1
1600.3.b.p 4 5.c odd 4 1
1600.3.b.p 4 20.e even 4 1
1600.3.h.o 8 1.a even 1 1 trivial
1600.3.h.o 8 4.b odd 2 1 inner
1600.3.h.o 8 5.b even 2 1 inner
1600.3.h.o 8 20.d odd 2 1 inner

Hecke kernels

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

\( T_{3}^{4} - 26T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{4} - 136T_{7}^{2} + 2000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 26 T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 136 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 130 T^{2} + 125)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 336 T^{2} + 25600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 346 T^{2} + 24025)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1370 T^{2} + 465125)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 776 T^{2} + 147920)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 20 T - 64)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 520 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2376 T^{2} + 739600)^{2} \) Copy content Toggle raw display
$41$ \( (T - 17)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 3056 T^{2} + 128000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4976 T^{2} + 5242880)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8136 T^{2} + 1392400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 13680 T^{2} + 41472000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 56 T - 3316)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 3586 T^{2} + 1915805)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 19120 T^{2} + 67712000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 6826 T^{2} + 9517225)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 7720 T^{2} + 6962000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 5426 T^{2} + 3621005)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 110 T + 2861)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 23976 T^{2} + 32400)^{2} \) Copy content Toggle raw display
show more
show less