Properties

Label 1600.4.a.cw
Level $1600$
Weight $4$
Character orbit 1600.a
Self dual yes
Analytic conductor $94.403$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,4,Mod(1,1600)]
 
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([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)

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: yes
Analytic conductor: \(94.4030560092\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2106005.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 24x^{2} + 25x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 800)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} - \beta_1) q^{7} + (\beta_{2} + 24) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} - \beta_1) q^{7} + (\beta_{2} + 24) q^{9} + ( - 3 \beta_{3} - 4 \beta_1) q^{11} + ( - \beta_{2} - 18) q^{13} + ( - \beta_{2} - 15) q^{17} + (2 \beta_{3} + 5 \beta_1) q^{19} + ( - \beta_{2} - 56) q^{21} + (6 \beta_{3} - 16 \beta_1) q^{23} + (\beta_{3} + 48 \beta_1) q^{27} + (3 \beta_{2} + 18) q^{29} + ( - 10 \beta_{3} + 16 \beta_1) q^{31} + ( - 4 \beta_{2} - 189) q^{33} + ( - 3 \beta_{2} - 232) q^{37} + ( - \beta_{3} - 69 \beta_1) q^{39} + ( - 4 \beta_{2} + 181) q^{41} + (11 \beta_{3} - 13 \beta_1) q^{43} + ( - 11 \beta_{3} - 31 \beta_1) q^{47} + ( - 4 \beta_{2} - 27) q^{49} + ( - \beta_{3} - 66 \beta_1) q^{51} + 222 q^{53} + (5 \beta_{2} + 245) q^{57} + ( - 25 \beta_{3} - 53 \beta_1) q^{59} + (\beta_{2} - 444) q^{61} + ( - 28 \beta_{3} - 80 \beta_1) q^{63} + ( - 25 \beta_{3} + 46 \beta_1) q^{67} + ( - 16 \beta_{2} - 846) q^{69} + (29 \beta_{3} + \beta_1) q^{71} + (17 \beta_{2} - 265) q^{73} + (19 \beta_{2} - 556) q^{77} + ( - 9 \beta_{3} - 107 \beta_1) q^{79} + (21 \beta_{2} + 1795) q^{81} + ( - 37 \beta_{3} - 114 \beta_1) q^{83} + (3 \beta_{3} + 171 \beta_1) q^{87} + (5 \beta_{2} + 225) q^{89} + (34 \beta_{3} + 74 \beta_1) q^{91} + (16 \beta_{2} + 866) q^{93} + ( - 10 \beta_{2} - 126) q^{97} + (77 \beta_{3} - 285 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 96 q^{9} - 72 q^{13} - 60 q^{17} - 224 q^{21} + 72 q^{29} - 756 q^{33} - 928 q^{37} + 724 q^{41} - 108 q^{49} + 888 q^{53} + 980 q^{57} - 1776 q^{61} - 3384 q^{69} - 1060 q^{73} - 2224 q^{77} + 7180 q^{81} + 900 q^{89} + 3464 q^{93} - 504 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 24x^{2} + 25x - 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 4\nu - 50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{3} - 12\nu^{2} - 198\nu + 101 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 + 52 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 3\beta_{2} + 105\beta _1 + 154 ) / 8 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.54854
0.389272
0.610728
5.54854
0 −10.0971 0 0 0 10.5923 0 74.9510 0
1.2 0 −0.221457 0 0 0 22.7992 0 −26.9510 0
1.3 0 0.221457 0 0 0 −22.7992 0 −26.9510 0
1.4 0 10.0971 0 0 0 −10.5923 0 74.9510 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.4.a.cw 4
4.b odd 2 1 inner 1600.4.a.cw 4
5.b even 2 1 1600.4.a.cx 4
8.b even 2 1 800.4.a.bb yes 4
8.d odd 2 1 800.4.a.bb yes 4
20.d odd 2 1 1600.4.a.cx 4
40.e odd 2 1 800.4.a.ba 4
40.f even 2 1 800.4.a.ba 4
40.i odd 4 2 800.4.c.o 8
40.k even 4 2 800.4.c.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.4.a.ba 4 40.e odd 2 1
800.4.a.ba 4 40.f even 2 1
800.4.a.bb yes 4 8.b even 2 1
800.4.a.bb yes 4 8.d odd 2 1
800.4.c.o 8 40.i odd 4 2
800.4.c.o 8 40.k even 4 2
1600.4.a.cw 4 1.a even 1 1 trivial
1600.4.a.cw 4 4.b odd 2 1 inner
1600.4.a.cx 4 5.b even 2 1
1600.4.a.cx 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\):

\( T_{3}^{4} - 102T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{4} - 632T_{7}^{2} + 58320 \) Copy content Toggle raw display
\( T_{11}^{4} - 5982T_{11}^{2} + 6762845 \) Copy content Toggle raw display
\( T_{13}^{2} + 36T_{13} - 2272 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 102T^{2} + 5 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 632 T^{2} + 58320 \) Copy content Toggle raw display
$11$ \( T^{4} - 5982 T^{2} + \cdots + 6762845 \) Copy content Toggle raw display
$13$ \( (T^{2} + 36 T - 2272)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 30 T - 2371)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 4390 T^{2} + \cdots + 4753125 \) Copy content Toggle raw display
$23$ \( T^{4} - 46392 T^{2} + \cdots + 523059920 \) Copy content Toggle raw display
$29$ \( (T^{2} - 36 T - 23040)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 80312 T^{2} + \cdots + 1457948880 \) Copy content Toggle raw display
$37$ \( (T^{2} + 464 T + 30460)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 362 T - 8775)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 81808 T^{2} + \cdots + 1179648000 \) Copy content Toggle raw display
$47$ \( T^{4} - 152912 T^{2} + \cdots + 5516513280 \) Copy content Toggle raw display
$53$ \( (T - 222)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 578768 T^{2} + \cdots + 83483873280 \) Copy content Toggle raw display
$61$ \( (T^{2} + 888 T + 194540)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 557582 T^{2} + \cdots + 75081483405 \) Copy content Toggle raw display
$71$ \( T^{4} - 428432 T^{2} + \cdots + 7787520 \) Copy content Toggle raw display
$73$ \( (T^{2} + 530 T - 680019)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 1189848 T^{2} + \cdots + 37300611920 \) Copy content Toggle raw display
$83$ \( T^{4} - 1939422 T^{2} + \cdots + 842120280125 \) Copy content Toggle raw display
$89$ \( (T^{2} - 450 T - 14275)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 252 T - 243724)^{2} \) Copy content Toggle raw display
show more
show less