Properties

Label 1600.2.c.a
Level $1600$
Weight $2$
Character orbit 1600.c
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(449,1600)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.c (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: \(12.7760643234\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} - 2 i q^{7} - 6 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} - 2 i q^{7} - 6 q^{9} - q^{11} - 4 i q^{13} - 5 i q^{17} + q^{19} + 6 q^{21} - 2 i q^{23} - 9 i q^{27} - 8 q^{29} + 10 q^{31} - 3 i q^{33} - 6 i q^{37} + 12 q^{39} - 3 q^{41} - 4 i q^{43} - 4 i q^{47} + 3 q^{49} + 15 q^{51} - 6 i q^{53} + 3 i q^{57} + 8 q^{59} - 10 q^{61} + 12 i q^{63} - i q^{67} + 6 q^{69} - 12 q^{71} + 3 i q^{73} + 2 i q^{77} - 6 q^{79} + 9 q^{81} + 13 i q^{83} - 24 i q^{87} + 9 q^{89} - 8 q^{91} + 30 i q^{93} + 14 i q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{9} - 2 q^{11} + 2 q^{19} + 12 q^{21} - 16 q^{29} + 20 q^{31} + 24 q^{39} - 6 q^{41} + 6 q^{49} + 30 q^{51} + 16 q^{59} - 20 q^{61} + 12 q^{69} - 24 q^{71} - 12 q^{79} + 18 q^{81} + 18 q^{89} - 16 q^{91} + 12 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 3.00000i 0 0 0 2.00000i 0 −6.00000 0
449.2 0 3.00000i 0 0 0 2.00000i 0 −6.00000 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
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 1600.2.c.a 2
4.b odd 2 1 1600.2.c.b 2
5.b even 2 1 inner 1600.2.c.a 2
5.c odd 4 1 1600.2.a.a 1
5.c odd 4 1 1600.2.a.x 1
8.b even 2 1 200.2.c.a 2
8.d odd 2 1 400.2.c.a 2
20.d odd 2 1 1600.2.c.b 2
20.e even 4 1 1600.2.a.b 1
20.e even 4 1 1600.2.a.y 1
24.f even 2 1 3600.2.f.n 2
24.h odd 2 1 1800.2.f.f 2
40.e odd 2 1 400.2.c.a 2
40.f even 2 1 200.2.c.a 2
40.i odd 4 1 200.2.a.a 1
40.i odd 4 1 200.2.a.e yes 1
40.k even 4 1 400.2.a.a 1
40.k even 4 1 400.2.a.h 1
120.i odd 2 1 1800.2.f.f 2
120.m even 2 1 3600.2.f.n 2
120.q odd 4 1 3600.2.a.m 1
120.q odd 4 1 3600.2.a.bf 1
120.w even 4 1 1800.2.a.h 1
120.w even 4 1 1800.2.a.r 1
280.s even 4 1 9800.2.a.c 1
280.s even 4 1 9800.2.a.bq 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.a.a 1 40.i odd 4 1
200.2.a.e yes 1 40.i odd 4 1
200.2.c.a 2 8.b even 2 1
200.2.c.a 2 40.f even 2 1
400.2.a.a 1 40.k even 4 1
400.2.a.h 1 40.k even 4 1
400.2.c.a 2 8.d odd 2 1
400.2.c.a 2 40.e odd 2 1
1600.2.a.a 1 5.c odd 4 1
1600.2.a.b 1 20.e even 4 1
1600.2.a.x 1 5.c odd 4 1
1600.2.a.y 1 20.e even 4 1
1600.2.c.a 2 1.a even 1 1 trivial
1600.2.c.a 2 5.b even 2 1 inner
1600.2.c.b 2 4.b odd 2 1
1600.2.c.b 2 20.d odd 2 1
1800.2.a.h 1 120.w even 4 1
1800.2.a.r 1 120.w even 4 1
1800.2.f.f 2 24.h odd 2 1
1800.2.f.f 2 120.i odd 2 1
3600.2.a.m 1 120.q odd 4 1
3600.2.a.bf 1 120.q odd 4 1
3600.2.f.n 2 24.f even 2 1
3600.2.f.n 2 120.m even 2 1
9800.2.a.c 1 280.s even 4 1
9800.2.a.bq 1 280.s even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display
\( T_{19} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 25 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 8)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 9 \) Copy content Toggle raw display
$79$ \( (T + 6)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 169 \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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