Properties

Label 1600.3.b.f
Level $1600$
Weight $3$
Character orbit 1600.b
Analytic conductor $43.597$
Analytic rank $0$
Dimension $2$
CM discriminant -20
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(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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 4i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - \beta q^{7} - 7 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - \beta q^{7} - 7 q^{9} + 16 q^{21} - 11 \beta q^{23} + 2 \beta q^{27} - 22 q^{29} + 62 q^{41} - 19 \beta q^{43} - \beta q^{47} + 33 q^{49} + 58 q^{61} + 7 \beta q^{63} - 29 \beta q^{67} + 176 q^{69} - 95 q^{81} - 19 \beta q^{83} - 22 \beta q^{87} + 142 q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{9} + 32 q^{21} - 44 q^{29} + 124 q^{41} + 66 q^{49} + 116 q^{61} + 352 q^{69} - 190 q^{81} + 284 q^{89}+O(q^{100}) \) 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
1.00000i
1.00000i
0 4.00000i 0 0 0 4.00000i 0 −7.00000 0
1151.2 0 4.00000i 0 0 0 4.00000i 0 −7.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 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.f 2
4.b odd 2 1 inner 1600.3.b.f 2
5.b even 2 1 inner 1600.3.b.f 2
5.c odd 4 1 320.3.h.a 1
5.c odd 4 1 320.3.h.b 1
8.b even 2 1 100.3.b.c 2
8.d odd 2 1 100.3.b.c 2
20.d odd 2 1 CM 1600.3.b.f 2
20.e even 4 1 320.3.h.a 1
20.e even 4 1 320.3.h.b 1
24.f even 2 1 900.3.c.h 2
24.h odd 2 1 900.3.c.h 2
40.e odd 2 1 100.3.b.c 2
40.f even 2 1 100.3.b.c 2
40.i odd 4 1 20.3.d.a 1
40.i odd 4 1 20.3.d.b yes 1
40.k even 4 1 20.3.d.a 1
40.k even 4 1 20.3.d.b yes 1
80.i odd 4 1 1280.3.e.b 2
80.i odd 4 1 1280.3.e.c 2
80.j even 4 1 1280.3.e.b 2
80.j even 4 1 1280.3.e.c 2
80.s even 4 1 1280.3.e.b 2
80.s even 4 1 1280.3.e.c 2
80.t odd 4 1 1280.3.e.b 2
80.t odd 4 1 1280.3.e.c 2
120.i odd 2 1 900.3.c.h 2
120.m even 2 1 900.3.c.h 2
120.q odd 4 1 180.3.f.a 1
120.q odd 4 1 180.3.f.b 1
120.w even 4 1 180.3.f.a 1
120.w even 4 1 180.3.f.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 40.i odd 4 1
20.3.d.a 1 40.k even 4 1
20.3.d.b yes 1 40.i odd 4 1
20.3.d.b yes 1 40.k even 4 1
100.3.b.c 2 8.b even 2 1
100.3.b.c 2 8.d odd 2 1
100.3.b.c 2 40.e odd 2 1
100.3.b.c 2 40.f even 2 1
180.3.f.a 1 120.q odd 4 1
180.3.f.a 1 120.w even 4 1
180.3.f.b 1 120.q odd 4 1
180.3.f.b 1 120.w even 4 1
320.3.h.a 1 5.c odd 4 1
320.3.h.a 1 20.e even 4 1
320.3.h.b 1 5.c odd 4 1
320.3.h.b 1 20.e even 4 1
900.3.c.h 2 24.f even 2 1
900.3.c.h 2 24.h odd 2 1
900.3.c.h 2 120.i odd 2 1
900.3.c.h 2 120.m even 2 1
1280.3.e.b 2 80.i odd 4 1
1280.3.e.b 2 80.j even 4 1
1280.3.e.b 2 80.s even 4 1
1280.3.e.b 2 80.t odd 4 1
1280.3.e.c 2 80.i odd 4 1
1280.3.e.c 2 80.j even 4 1
1280.3.e.c 2 80.s even 4 1
1280.3.e.c 2 80.t odd 4 1
1600.3.b.f 2 1.a even 1 1 trivial
1600.3.b.f 2 4.b odd 2 1 inner
1600.3.b.f 2 5.b even 2 1 inner
1600.3.b.f 2 20.d odd 2 1 CM

Hecke kernels

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

\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1936 \) Copy content Toggle raw display
$29$ \( (T + 22)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T - 62)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 5776 \) Copy content Toggle raw display
$47$ \( T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 58)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 13456 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5776 \) Copy content Toggle raw display
$89$ \( (T - 142)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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