Properties

Label 1600.3.b.p
Level $1600$
Weight $3$
Character orbit 1600.b
Analytic conductor $43.597$
Analytic rank $0$
Dimension $4$
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] = [1600,3,Mod(1151,1600)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.b (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: \(4\)
Coefficient field: 4.0.8405.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 8x^{2} - 7x + 2 \) 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 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{2} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{2} - 4) q^{9} - \beta_{3} q^{11} + ( - \beta_{2} + 2) q^{13} + ( - \beta_{2} - 3) q^{17} + ( - 2 \beta_{3} - 5 \beta_1) q^{19} + ( - \beta_{2} + 8) q^{21} + (2 \beta_{3} - 4 \beta_1) q^{23} + ( - \beta_{3} + 8 \beta_1) q^{27} + ( - \beta_{2} - 10) q^{29} - 2 \beta_{3} q^{31} - 5 q^{33} + (\beta_{2} - 32) q^{37} + (\beta_{3} - 15 \beta_1) q^{39} + 17 q^{41} + (\beta_{3} - 11 \beta_1) q^{43} + (3 \beta_{3} + 11 \beta_1) q^{47} + ( - 4 \beta_{2} - 19) q^{49} + (\beta_{3} - 10 \beta_1) q^{51} + (4 \beta_{2} - 38) q^{53} + (5 \beta_{2} - 75) q^{57} + (9 \beta_{3} - 15 \beta_1) q^{59} + (5 \beta_{2} + 28) q^{61} + ( - 8 \beta_{3} - 12 \beta_1) q^{63} + (5 \beta_{3} - 6 \beta_1) q^{67} + (4 \beta_{2} - 42) q^{69} + (7 \beta_{3} - 25 \beta_1) q^{71} + (\beta_{2} - 57) q^{73} + ( - 5 \beta_{2} - 60) q^{77} + ( - 7 \beta_{3} - 5 \beta_1) q^{79} + (\beta_{2} + 63) q^{81} + ( - 3 \beta_{3} + 14 \beta_1) q^{83} + (\beta_{3} - 3 \beta_1) q^{87} + (\beta_{2} - 55) q^{89} + (10 \beta_{3} + 10 \beta_1) q^{91} - 10 q^{93} + ( - 6 \beta_{2} - 78) q^{97} + ( - 9 \beta_{3} + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{9} + 8 q^{13} - 12 q^{17} + 32 q^{21} - 40 q^{29} - 20 q^{33} - 128 q^{37} + 68 q^{41} - 76 q^{49} - 152 q^{53} - 300 q^{57} + 112 q^{61} - 168 q^{69} - 228 q^{73} - 240 q^{77} + 252 q^{81} - 220 q^{89} - 40 q^{93} - 312 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 4\nu + 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{3} - 12\nu^{2} + 58\nu - 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 3\beta_{2} - 23\beta _1 - 38 ) / 8 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
0.500000 + 2.53999i
0.500000 + 0.220086i
0.500000 0.220086i
0.500000 2.53999i
0 5.07999i 0 0 0 4.09573i 0 −16.8062 0
1151.2 0 0.440172i 0 0 0 10.9190i 0 8.80625 0
1151.3 0 0.440172i 0 0 0 10.9190i 0 8.80625 0
1151.4 0 5.07999i 0 0 0 4.09573i 0 −16.8062 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
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 1600.3.b.p 4
4.b odd 2 1 inner 1600.3.b.p 4
5.b even 2 1 1600.3.b.o 4
5.c odd 4 2 1600.3.h.o 8
8.b even 2 1 100.3.b.d 4
8.d odd 2 1 100.3.b.d 4
20.d odd 2 1 1600.3.b.o 4
20.e even 4 2 1600.3.h.o 8
24.f even 2 1 900.3.c.m 4
24.h odd 2 1 900.3.c.m 4
40.e odd 2 1 100.3.b.e yes 4
40.f even 2 1 100.3.b.e yes 4
40.i odd 4 2 100.3.d.a 8
40.k even 4 2 100.3.d.a 8
120.i odd 2 1 900.3.c.l 4
120.m even 2 1 900.3.c.l 4
120.q odd 4 2 900.3.f.d 8
120.w even 4 2 900.3.f.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.3.b.d 4 8.b even 2 1
100.3.b.d 4 8.d odd 2 1
100.3.b.e yes 4 40.e odd 2 1
100.3.b.e yes 4 40.f even 2 1
100.3.d.a 8 40.i odd 4 2
100.3.d.a 8 40.k even 4 2
900.3.c.l 4 120.i odd 2 1
900.3.c.l 4 120.m even 2 1
900.3.c.m 4 24.f even 2 1
900.3.c.m 4 24.h odd 2 1
900.3.f.d 8 120.q odd 4 2
900.3.f.d 8 120.w even 4 2
1600.3.b.o 4 5.b even 2 1
1600.3.b.o 4 20.d odd 2 1
1600.3.b.p 4 1.a even 1 1 trivial
1600.3.b.p 4 4.b odd 2 1 inner
1600.3.h.o 8 5.c odd 4 2
1600.3.h.o 8 20.e even 4 2

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
\( T_{13}^{2} - 4T_{13} - 160 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 26T^{2} + 5 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 136T^{2} + 2000 \) Copy content Toggle raw display
$11$ \( T^{4} + 130T^{2} + 125 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 160)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 155)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 1370 T^{2} + 465125 \) Copy content Toggle raw display
$23$ \( T^{4} + 776 T^{2} + 147920 \) Copy content Toggle raw display
$29$ \( (T^{2} + 20 T - 64)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 520T^{2} + 2000 \) Copy content Toggle raw display
$37$ \( (T^{2} + 64 T + 860)^{2} \) Copy content Toggle raw display
$41$ \( (T - 17)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 3056 T^{2} + 128000 \) Copy content Toggle raw display
$47$ \( T^{4} + 4976 T^{2} + \cdots + 5242880 \) Copy content Toggle raw display
$53$ \( (T^{2} + 76 T - 1180)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 13680 T^{2} + \cdots + 41472000 \) Copy content Toggle raw display
$61$ \( (T^{2} - 56 T - 3316)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 3586 T^{2} + \cdots + 1915805 \) Copy content Toggle raw display
$71$ \( T^{4} + 19120 T^{2} + \cdots + 67712000 \) Copy content Toggle raw display
$73$ \( (T^{2} + 114 T + 3085)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 7720 T^{2} + \cdots + 6962000 \) Copy content Toggle raw display
$83$ \( T^{4} + 5426 T^{2} + \cdots + 3621005 \) Copy content Toggle raw display
$89$ \( (T^{2} + 110 T + 2861)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 156 T + 180)^{2} \) Copy content Toggle raw display
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