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Properties

Label 40.5.l.c
Level $40$
Weight $5$
Character orbit 40.l
Analytic conductor $4.135$
Analytic rank $0$
Dimension $6$
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] = [40,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.17");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(4.13479852335\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.313431616.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 27x^{4} + 145x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1 - 1) q^{3} + ( - \beta_{5} + \beta_{3} + 7 \beta_1 - 1) q^{5} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} + \cdots - 6) q^{7}+ \cdots + ( - 4 \beta_{5} + 5 \beta_{3} + \cdots + 49 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1 - 1) q^{3} + ( - \beta_{5} + \beta_{3} + 7 \beta_1 - 1) q^{5} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} + \cdots - 6) q^{7}+ \cdots + ( - 78 \beta_{5} - 25 \beta_{3} + \cdots - 3524 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{3} - 4 q^{5} - 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{3} - 4 q^{5} - 40 q^{7} + 72 q^{11} - 62 q^{13} - 280 q^{15} + 418 q^{17} + 1736 q^{21} - 1760 q^{23} - 2974 q^{25} + 2248 q^{27} + 2864 q^{31} - 8 q^{33} - 4224 q^{35} + 5366 q^{37} + 1656 q^{41} - 2760 q^{43} - 14678 q^{45} - 432 q^{47} + 15840 q^{51} + 534 q^{53} - 136 q^{55} + 12576 q^{57} + 216 q^{61} - 19552 q^{63} - 15082 q^{65} - 1336 q^{67} + 2512 q^{71} + 20230 q^{73} + 22240 q^{75} + 5832 q^{77} - 16526 q^{81} - 22736 q^{83} - 6798 q^{85} - 22608 q^{87} + 12848 q^{91} + 36520 q^{93} + 33064 q^{95} - 9418 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 27x^{4} + 145x^{2} + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 39\nu^{3} + 313\nu ) / 300 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{5} + 60\nu^{4} - 441\nu^{3} + 1140\nu^{2} - 547\nu + 1380 ) / 300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -19\nu^{5} - 60\nu^{4} - 441\nu^{3} - 1140\nu^{2} - 547\nu - 1380 ) / 300 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{4} + 58\nu^{2} + 231 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41\nu^{5} + 999\nu^{3} + 4033\nu ) / 100 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + 3\beta_{3} + 3\beta_{2} - 9\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} - 37 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{5} - 22\beta_{3} - 22\beta_{2} + 271\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -19\beta_{4} - 29\beta_{3} + 29\beta_{2} + 611 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 389\beta_{5} + 777\beta_{3} + 777\beta_{2} - 12321\beta_1 ) / 20 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
− 2.35597i
− 1.13432i
4.49029i
2.35597i
1.13432i
− 4.49029i
0 −11.3304 + 11.3304i 0 9.33042 − 23.1936i 0 −42.8544 − 42.8544i 0 − 175.757i 0
17.2 0 −0.958314 + 0.958314i 0 −1.04169 + 24.9783i 0 26.0617 + 26.0617i 0 79.1633i 0
17.3 0 8.28874 − 8.28874i 0 −10.2887 − 22.7847i 0 −3.20721 − 3.20721i 0 − 56.4063i 0
33.1 0 −11.3304 − 11.3304i 0 9.33042 + 23.1936i 0 −42.8544 + 42.8544i 0 175.757i 0
33.2 0 −0.958314 − 0.958314i 0 −1.04169 − 24.9783i 0 26.0617 − 26.0617i 0 − 79.1633i 0
33.3 0 8.28874 + 8.28874i 0 −10.2887 + 22.7847i 0 −3.20721 + 3.20721i 0 56.4063i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.3
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 40.5.l.c ✓ 6
3.b odd 2 1 360.5.v.c 6
4.b odd 2 1 80.5.p.g 6
5.b even 2 1 200.5.l.e 6
5.c odd 4 1 inner 40.5.l.c ✓ 6
5.c odd 4 1 200.5.l.e 6
8.b even 2 1 320.5.p.p 6
8.d odd 2 1 320.5.p.o 6
15.e even 4 1 360.5.v.c 6
20.d odd 2 1 400.5.p.o 6
20.e even 4 1 80.5.p.g 6
20.e even 4 1 400.5.p.o 6
40.i odd 4 1 320.5.p.p 6
40.k even 4 1 320.5.p.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.5.l.c ✓ 6 1.a even 1 1 trivial
40.5.l.c ✓ 6 5.c odd 4 1 inner
80.5.p.g 6 4.b odd 2 1
80.5.p.g 6 20.e even 4 1
200.5.l.e 6 5.b even 2 1
200.5.l.e 6 5.c odd 4 1
320.5.p.o 6 8.d odd 2 1
320.5.p.o 6 40.k even 4 1
320.5.p.p 6 8.b even 2 1
320.5.p.p 6 40.i odd 4 1
360.5.v.c 6 3.b odd 2 1
360.5.v.c 6 15.e even 4 1
400.5.p.o 6 20.d odd 2 1
400.5.p.o 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 8T_{3}^{5} + 32T_{3}^{4} - 1096T_{3}^{3} + 33124T_{3}^{2} + 65520T_{3} + 64800 \) acting on \(S_{5}^{\mathrm{new}}(40, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{5} + \cdots + 64800 \) Copy content Toggle raw display
$5$ \( T^{6} + 4 T^{5} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{6} + 40 T^{5} + \cdots + 102645792 \) Copy content Toggle raw display
$11$ \( (T^{3} - 36 T^{2} + \cdots - 30304)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 87454385847432 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 86815346805000 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 58\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{3} - 1432 T^{2} + \cdots + 646120400)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 25\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( (T^{3} - 828 T^{2} + \cdots - 281777056)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{3} - 108 T^{2} + \cdots - 3747343200)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( (T^{3} - 1256 T^{2} + \cdots + 168954075280)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 36\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 57\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 59\!\cdots\!32 \) Copy content Toggle raw display
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