Properties

Label 40.3.l.b
Level $40$
Weight $3$
Character orbit 40.l
Analytic conductor $1.090$
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] = [40,3,Mod(17,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 40.l (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: \(1.08992105744\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + (\beta_{3} - 2 \beta_1 - 1) q^{5} + ( - \beta_{3} - 3 \beta_1 + 3) q^{7} + ( - \beta_{3} - \beta_{2} + 12 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} + (\beta_{3} - 2 \beta_1 - 1) q^{5} + ( - \beta_{3} - 3 \beta_1 + 3) q^{7} + ( - \beta_{3} - \beta_{2} + 12 \beta_1) q^{9} + (\beta_{3} - \beta_{2} - \beta_1 + 6) q^{11} + ( - 2 \beta_{2} - 9 \beta_1 - 7) q^{13} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 21) q^{15} + (2 \beta_{3} - 5 \beta_1 + 5) q^{17} + (2 \beta_{3} + 2 \beta_{2} + 10 \beta_1) q^{19} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 14) q^{21} + ( - 3 \beta_{2} + 8 \beta_1 + 11) q^{23} + ( - 3 \beta_{3} + 3 \beta_{2} - 13 \beta_1 - 3) q^{25} + (4 \beta_{3} - 24 \beta_1 + 24) q^{27} + (4 \beta_{3} + 4 \beta_{2} + 20 \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} + \beta_1 - 10) q^{31} + (6 \beta_{2} - 20 \beta_1 - 26) q^{33} + (5 \beta_{3} + 2 \beta_{2} + 19 \beta_1 - 9) q^{35} + ( - 8 \beta_{3} + 19 \beta_1 - 19) q^{37} + ( - 7 \beta_{3} - 7 \beta_{2} - 33 \beta_1) q^{39} + (9 \beta_{3} - 9 \beta_{2} - 9 \beta_1 + 22) q^{41} + (\beta_{2} + 24 \beta_1 + 23) q^{43} + (3 \beta_{3} - 12 \beta_{2} - 5 \beta_1 + 46) q^{45} + ( - 5 \beta_{3} + 29 \beta_1 - 29) q^{47} + ( - 7 \beta_{3} - 7 \beta_{2} + 4 \beta_1) q^{49} + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 50) q^{51} + ( - 3 \beta_1 - 3) q^{53} + (5 \beta_{3} + 3 \beta_{2} - 29 \beta_1 + 14) q^{55} + (8 \beta_{3} + 32 \beta_1 - 32) q^{57} + (2 \beta_{3} + 2 \beta_{2} - 54 \beta_1) q^{59} + (\beta_{3} - \beta_{2} - \beta_1 - 50) q^{61} + (5 \beta_{2} + 24 \beta_1 + 19) q^{63} + ( - 5 \beta_{3} + 11 \beta_{2} + 32 \beta_1 + 33) q^{65} + ( - \beta_{3} - 31 \beta_1 + 31) q^{67} + (11 \beta_{3} + 11 \beta_{2} - 71 \beta_1) q^{69} + ( - \beta_{3} + \beta_{2} + \beta_1 + 38) q^{71} + ( - 8 \beta_{2} + 33 \beta_1 + 41) q^{73} + ( - 16 \beta_{3} - 3 \beta_{2} + 76 \beta_1 + 47) q^{75} + (2 \beta_{3} + 2 \beta_1 - 2) q^{77} + (16 \beta_{3} + 16 \beta_{2} - 16 \beta_1) q^{79} + ( - 15 \beta_{3} + 15 \beta_{2} + 15 \beta_1 - 11) q^{81} + ( - 11 \beta_{2} - 52 \beta_1 - 41) q^{83} + (\beta_{3} + 7 \beta_{2} - 38 \beta_1 - 15) q^{85} + (16 \beta_{3} + 64 \beta_1 - 64) q^{87} + ( - 8 \beta_{3} - 8 \beta_{2} - 40 \beta_1) q^{89} + (15 \beta_{3} - 15 \beta_{2} - 15 \beta_1 - 82) q^{91} + ( - 10 \beta_{2} + 20 \beta_1 + 30) q^{93} + ( - 6 \beta_{3} - 10 \beta_{2} - 58 \beta_1 - 24) q^{95} + ( - 49 \beta_1 + 49) q^{97} + ( - 17 \beta_{3} - 17 \beta_{2} + 101 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 6 q^{5} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 6 q^{5} + 14 q^{7} + 20 q^{11} - 32 q^{13} - 82 q^{15} + 16 q^{17} + 68 q^{21} + 38 q^{23} + 88 q^{27} - 36 q^{31} - 92 q^{33} - 42 q^{35} - 60 q^{37} + 52 q^{41} + 94 q^{43} + 154 q^{45} - 106 q^{47} - 180 q^{51} - 12 q^{53} + 52 q^{55} - 144 q^{57} - 204 q^{61} + 86 q^{63} + 164 q^{65} + 126 q^{67} + 156 q^{71} + 148 q^{73} + 214 q^{75} - 12 q^{77} + 16 q^{81} - 186 q^{83} - 48 q^{85} - 288 q^{87} - 388 q^{91} + 100 q^{93} - 104 q^{95} + 196 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 21x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 11\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu + 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 10\nu^{2} + 21\nu - 110 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + \beta _1 - 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{3} - 11\beta_{2} + 31\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
3.70156i
2.70156i
3.70156i
2.70156i
0 −3.70156 + 3.70156i 0 1.70156 + 4.70156i 0 0.298438 + 0.298438i 0 18.4031i 0
17.2 0 2.70156 2.70156i 0 −4.70156 1.70156i 0 6.70156 + 6.70156i 0 5.59688i 0
33.1 0 −3.70156 3.70156i 0 1.70156 4.70156i 0 0.298438 0.298438i 0 18.4031i 0
33.2 0 2.70156 + 2.70156i 0 −4.70156 + 1.70156i 0 6.70156 6.70156i 0 5.59688i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.3.l.b 4
3.b odd 2 1 360.3.v.c 4
4.b odd 2 1 80.3.p.d 4
5.b even 2 1 200.3.l.e 4
5.c odd 4 1 inner 40.3.l.b 4
5.c odd 4 1 200.3.l.e 4
8.b even 2 1 320.3.p.l 4
8.d odd 2 1 320.3.p.i 4
12.b even 2 1 720.3.bh.l 4
15.d odd 2 1 1800.3.v.k 4
15.e even 4 1 360.3.v.c 4
15.e even 4 1 1800.3.v.k 4
20.d odd 2 1 400.3.p.i 4
20.e even 4 1 80.3.p.d 4
20.e even 4 1 400.3.p.i 4
40.i odd 4 1 320.3.p.l 4
40.k even 4 1 320.3.p.i 4
60.l odd 4 1 720.3.bh.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.l.b 4 1.a even 1 1 trivial
40.3.l.b 4 5.c odd 4 1 inner
80.3.p.d 4 4.b odd 2 1
80.3.p.d 4 20.e even 4 1
200.3.l.e 4 5.b even 2 1
200.3.l.e 4 5.c odd 4 1
320.3.p.i 4 8.d odd 2 1
320.3.p.i 4 40.k even 4 1
320.3.p.l 4 8.b even 2 1
320.3.p.l 4 40.i odd 4 1
360.3.v.c 4 3.b odd 2 1
360.3.v.c 4 15.e even 4 1
400.3.p.i 4 20.d odd 2 1
400.3.p.i 4 20.e even 4 1
720.3.bh.l 4 12.b even 2 1
720.3.bh.l 4 60.l odd 4 1
1800.3.v.k 4 15.d odd 2 1
1800.3.v.k 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 40T_{3} + 400 \) acting on \(S_{3}^{\mathrm{new}}(40, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 2 T^{2} - 40 T + 400 \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + 18 T^{2} + 150 T + 625 \) Copy content Toggle raw display
$7$ \( T^{4} - 14 T^{3} + 98 T^{2} - 56 T + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 10 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 32 T^{3} + 512 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$17$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$19$ \( T^{4} + 528T^{2} + 4096 \) Copy content Toggle raw display
$23$ \( T^{4} - 38 T^{3} + 722 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{4} + 2112 T^{2} + 65536 \) Copy content Toggle raw display
$31$ \( (T^{2} + 18 T + 40)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 60 T^{3} + 1800 T^{2} + \cdots + 743044 \) Copy content Toggle raw display
$41$ \( (T^{2} - 26 T - 3152)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 94 T^{3} + 4418 T^{2} + \cdots + 1175056 \) Copy content Toggle raw display
$47$ \( T^{4} + 106 T^{3} + 5618 T^{2} + \cdots + 795664 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 6160 T^{2} + \cdots + 7573504 \) Copy content Toggle raw display
$61$ \( (T^{2} + 102 T + 2560)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 126 T^{3} + 7938 T^{2} + \cdots + 3857296 \) Copy content Toggle raw display
$71$ \( (T^{2} - 78 T + 1480)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 148 T^{3} + 10952 T^{2} + \cdots + 2033476 \) Copy content Toggle raw display
$79$ \( T^{4} + 21504 T^{2} + \cdots + 104857600 \) Copy content Toggle raw display
$83$ \( T^{4} + 186 T^{3} + 17298 T^{2} + \cdots + 3400336 \) Copy content Toggle raw display
$89$ \( T^{4} + 8448 T^{2} + \cdots + 1048576 \) Copy content Toggle raw display
$97$ \( (T^{2} - 98 T + 4802)^{2} \) Copy content Toggle raw display
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