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Properties

Label 3900.1.w.d
Level $3900$
Weight $1$
Character orbit 3900.w
Analytic conductor $1.946$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -260
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3900,1,Mod(2807,3900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3900.2807");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3900.w (of order \(4\), degree \(2\), 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: \(1.94635354927\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.9360.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{6} - \zeta_{8}^{3} q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} + q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{6} - \zeta_{8}^{3} q^{8} + q^{9} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} + \zeta_{8}^{2} q^{12} + \zeta_{8} q^{13} - q^{16} - \zeta_{8} q^{18} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{19} + (\zeta_{8}^{2} - 1) q^{22} + ( - \zeta_{8}^{2} + 1) q^{23} - \zeta_{8}^{3} q^{24} - \zeta_{8}^{2} q^{26} + q^{27} - q^{29} + (\zeta_{8}^{3} - \zeta_{8}) q^{31} + \zeta_{8} q^{32} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{33} + \zeta_{8}^{2} q^{36} + (\zeta_{8}^{2} - 1) q^{38} + \zeta_{8} q^{39} + (\zeta_{8}^{2} + 1) q^{43} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{44} + (\zeta_{8}^{3} - \zeta_{8}) q^{46} - q^{48} - \zeta_{8}^{2} q^{49} + \zeta_{8}^{3} q^{52} - \zeta_{8} q^{54} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{57} + 2 \zeta_{8} q^{58} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{59} + (\zeta_{8}^{2} + 1) q^{62} - \zeta_{8}^{2} q^{64} + (\zeta_{8}^{2} - 1) q^{66} + ( - \zeta_{8}^{2} + 1) q^{69} + (\zeta_{8}^{3} + \zeta_{8}) q^{71} - \zeta_{8}^{3} q^{72} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{76} - \zeta_{8}^{2} q^{78} + q^{81} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{86} - 2 q^{87} + ( - \zeta_{8}^{2} - 1) q^{88} + (\zeta_{8}^{2} + 1) q^{92} + (\zeta_{8}^{3} - \zeta_{8}) q^{93} + \zeta_{8} q^{96} + \zeta_{8}^{3} q^{97} + \zeta_{8}^{3} q^{98} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} - 4 q^{16} - 4 q^{22} + 4 q^{23} + 4 q^{27} - 8 q^{29} - 4 q^{38} + 4 q^{43} - 4 q^{48} + 4 q^{62} - 4 q^{66} + 4 q^{69} + 4 q^{81} - 8 q^{87} - 4 q^{88} + 4 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{8}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2807.1
0.707107 + 0.707107i
−0.707107 − 0.707107i
0.707107 − 0.707107i
−0.707107 + 0.707107i
−0.707107 − 0.707107i 1.00000 1.00000i 0 −0.707107 − 0.707107i 0 0.707107 − 0.707107i 1.00000 0
2807.2 0.707107 + 0.707107i 1.00000 1.00000i 0 0.707107 + 0.707107i 0 −0.707107 + 0.707107i 1.00000 0
3743.1 −0.707107 + 0.707107i 1.00000 − 1.00000i 0 −0.707107 + 0.707107i 0 0.707107 + 0.707107i 1.00000 0
3743.2 0.707107 − 0.707107i 1.00000 − 1.00000i 0 0.707107 − 0.707107i 0 −0.707107 − 0.707107i 1.00000 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
260.g odd 2 1 CM by \(\Q(\sqrt{-65}) \)
13.b even 2 1 inner
15.e even 4 1 inner
20.d odd 2 1 inner
60.l odd 4 1 inner
195.s even 4 1 inner
780.w odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3900.1.w.d yes 4
3.b odd 2 1 3900.1.w.b yes 4
4.b odd 2 1 3900.1.w.a ✓ 4
5.b even 2 1 3900.1.w.a ✓ 4
5.c odd 4 1 3900.1.w.b yes 4
5.c odd 4 1 3900.1.w.c yes 4
12.b even 2 1 3900.1.w.c yes 4
13.b even 2 1 inner 3900.1.w.d yes 4
15.d odd 2 1 3900.1.w.c yes 4
15.e even 4 1 3900.1.w.a ✓ 4
15.e even 4 1 inner 3900.1.w.d yes 4
20.d odd 2 1 inner 3900.1.w.d yes 4
20.e even 4 1 3900.1.w.b yes 4
20.e even 4 1 3900.1.w.c yes 4
39.d odd 2 1 3900.1.w.b yes 4
52.b odd 2 1 3900.1.w.a ✓ 4
60.h even 2 1 3900.1.w.b yes 4
60.l odd 4 1 3900.1.w.a ✓ 4
60.l odd 4 1 inner 3900.1.w.d yes 4
65.d even 2 1 3900.1.w.a ✓ 4
65.h odd 4 1 3900.1.w.b yes 4
65.h odd 4 1 3900.1.w.c yes 4
156.h even 2 1 3900.1.w.c yes 4
195.e odd 2 1 3900.1.w.c yes 4
195.s even 4 1 3900.1.w.a ✓ 4
195.s even 4 1 inner 3900.1.w.d yes 4
260.g odd 2 1 CM 3900.1.w.d yes 4
260.p even 4 1 3900.1.w.b yes 4
260.p even 4 1 3900.1.w.c yes 4
780.d even 2 1 3900.1.w.b yes 4
780.w odd 4 1 3900.1.w.a ✓ 4
780.w odd 4 1 inner 3900.1.w.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3900.1.w.a ✓ 4 4.b odd 2 1
3900.1.w.a ✓ 4 5.b even 2 1
3900.1.w.a ✓ 4 15.e even 4 1
3900.1.w.a ✓ 4 52.b odd 2 1
3900.1.w.a ✓ 4 60.l odd 4 1
3900.1.w.a ✓ 4 65.d even 2 1
3900.1.w.a ✓ 4 195.s even 4 1
3900.1.w.a ✓ 4 780.w odd 4 1
3900.1.w.b yes 4 3.b odd 2 1
3900.1.w.b yes 4 5.c odd 4 1
3900.1.w.b yes 4 20.e even 4 1
3900.1.w.b yes 4 39.d odd 2 1
3900.1.w.b yes 4 60.h even 2 1
3900.1.w.b yes 4 65.h odd 4 1
3900.1.w.b yes 4 260.p even 4 1
3900.1.w.b yes 4 780.d even 2 1
3900.1.w.c yes 4 5.c odd 4 1
3900.1.w.c yes 4 12.b even 2 1
3900.1.w.c yes 4 15.d odd 2 1
3900.1.w.c yes 4 20.e even 4 1
3900.1.w.c yes 4 65.h odd 4 1
3900.1.w.c yes 4 156.h even 2 1
3900.1.w.c yes 4 195.e odd 2 1
3900.1.w.c yes 4 260.p even 4 1
3900.1.w.d yes 4 1.a even 1 1 trivial
3900.1.w.d yes 4 13.b even 2 1 inner
3900.1.w.d yes 4 15.e even 4 1 inner
3900.1.w.d yes 4 20.d odd 2 1 inner
3900.1.w.d yes 4 60.l odd 4 1 inner
3900.1.w.d yes 4 195.s even 4 1 inner
3900.1.w.d yes 4 260.g odd 2 1 CM
3900.1.w.d yes 4 780.w odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3900, [\chi])\):

\( T_{11}^{2} + 2 \) Copy content Toggle raw display
\( T_{23}^{2} - 2T_{23} + 2 \) Copy content Toggle raw display
\( T_{29} + 2 \) Copy content Toggle raw display
\( T_{43}^{2} - 2T_{43} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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