Properties

Label 40.2.c.a
Level $40$
Weight $2$
Character orbit 40.c
Analytic conductor $0.319$
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] = [40,2,Mod(9,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, 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("40.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 40.c (of order \(2\), degree \(1\), 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: \(0.319401608085\)
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: \( 2 \)
Twist minimal: yes
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( - \beta - 1) q^{5} - \beta q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + ( - \beta - 1) q^{5} - \beta q^{7} - q^{9} - 4 q^{11} + 2 \beta q^{13} + ( - \beta + 4) q^{15} + 4 q^{19} + 4 q^{21} - \beta q^{23} + (2 \beta - 3) q^{25} + 2 \beta q^{27} - 2 q^{29} - 4 \beta q^{33} + (\beta - 4) q^{35} - 2 \beta q^{37} - 8 q^{39} + 2 q^{41} - 3 \beta q^{43} + (\beta + 1) q^{45} + 3 \beta q^{47} + 3 q^{49} - 2 \beta q^{53} + (4 \beta + 4) q^{55} + 4 \beta q^{57} + 12 q^{59} - 10 q^{61} + \beta q^{63} + ( - 2 \beta + 8) q^{65} - 7 \beta q^{67} + 4 q^{69} + 8 q^{71} + 4 \beta q^{73} + ( - 3 \beta - 8) q^{75} + 4 \beta q^{77} - 16 q^{79} - 11 q^{81} + \beta q^{83} - 2 \beta q^{87} - 6 q^{89} + 8 q^{91} + ( - 4 \beta - 4) q^{95} - 8 \beta q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{9} - 8 q^{11} + 8 q^{15} + 8 q^{19} + 8 q^{21} - 6 q^{25} - 4 q^{29} - 8 q^{35} - 16 q^{39} + 4 q^{41} + 2 q^{45} + 6 q^{49} + 8 q^{55} + 24 q^{59} - 20 q^{61} + 16 q^{65} + 8 q^{69} + 16 q^{71} - 16 q^{75} - 32 q^{79} - 22 q^{81} - 12 q^{89} + 16 q^{91} - 8 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
1.00000i
1.00000i
0 2.00000i 0 −1.00000 + 2.00000i 0 2.00000i 0 −1.00000 0
9.2 0 2.00000i 0 −1.00000 2.00000i 0 2.00000i 0 −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
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 40.2.c.a 2
3.b odd 2 1 360.2.f.c 2
4.b odd 2 1 80.2.c.a 2
5.b even 2 1 inner 40.2.c.a 2
5.c odd 4 1 200.2.a.b 1
5.c odd 4 1 200.2.a.d 1
7.b odd 2 1 1960.2.g.b 2
8.b even 2 1 320.2.c.c 2
8.d odd 2 1 320.2.c.b 2
12.b even 2 1 720.2.f.e 2
15.d odd 2 1 360.2.f.c 2
15.e even 4 1 1800.2.a.j 1
15.e even 4 1 1800.2.a.s 1
16.e even 4 1 1280.2.f.a 2
16.e even 4 1 1280.2.f.f 2
16.f odd 4 1 1280.2.f.b 2
16.f odd 4 1 1280.2.f.e 2
20.d odd 2 1 80.2.c.a 2
20.e even 4 1 400.2.a.b 1
20.e even 4 1 400.2.a.g 1
24.f even 2 1 2880.2.f.i 2
24.h odd 2 1 2880.2.f.h 2
35.c odd 2 1 1960.2.g.b 2
35.f even 4 1 9800.2.a.d 1
35.f even 4 1 9800.2.a.bf 1
40.e odd 2 1 320.2.c.b 2
40.f even 2 1 320.2.c.c 2
40.i odd 4 1 1600.2.a.f 1
40.i odd 4 1 1600.2.a.v 1
40.k even 4 1 1600.2.a.d 1
40.k even 4 1 1600.2.a.u 1
60.h even 2 1 720.2.f.e 2
60.l odd 4 1 3600.2.a.k 1
60.l odd 4 1 3600.2.a.bb 1
80.k odd 4 1 1280.2.f.b 2
80.k odd 4 1 1280.2.f.e 2
80.q even 4 1 1280.2.f.a 2
80.q even 4 1 1280.2.f.f 2
120.i odd 2 1 2880.2.f.h 2
120.m even 2 1 2880.2.f.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 1.a even 1 1 trivial
40.2.c.a 2 5.b even 2 1 inner
80.2.c.a 2 4.b odd 2 1
80.2.c.a 2 20.d odd 2 1
200.2.a.b 1 5.c odd 4 1
200.2.a.d 1 5.c odd 4 1
320.2.c.b 2 8.d odd 2 1
320.2.c.b 2 40.e odd 2 1
320.2.c.c 2 8.b even 2 1
320.2.c.c 2 40.f even 2 1
360.2.f.c 2 3.b odd 2 1
360.2.f.c 2 15.d odd 2 1
400.2.a.b 1 20.e even 4 1
400.2.a.g 1 20.e even 4 1
720.2.f.e 2 12.b even 2 1
720.2.f.e 2 60.h even 2 1
1280.2.f.a 2 16.e even 4 1
1280.2.f.a 2 80.q even 4 1
1280.2.f.b 2 16.f odd 4 1
1280.2.f.b 2 80.k odd 4 1
1280.2.f.e 2 16.f odd 4 1
1280.2.f.e 2 80.k odd 4 1
1280.2.f.f 2 16.e even 4 1
1280.2.f.f 2 80.q even 4 1
1600.2.a.d 1 40.k even 4 1
1600.2.a.f 1 40.i odd 4 1
1600.2.a.u 1 40.k even 4 1
1600.2.a.v 1 40.i odd 4 1
1800.2.a.j 1 15.e even 4 1
1800.2.a.s 1 15.e even 4 1
1960.2.g.b 2 7.b odd 2 1
1960.2.g.b 2 35.c odd 2 1
2880.2.f.h 2 24.h odd 2 1
2880.2.f.h 2 120.i odd 2 1
2880.2.f.i 2 24.f even 2 1
2880.2.f.i 2 120.m even 2 1
3600.2.a.k 1 60.l odd 4 1
3600.2.a.bb 1 60.l odd 4 1
9800.2.a.d 1 35.f even 4 1
9800.2.a.bf 1 35.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 256 \) Copy content Toggle raw display
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