Properties

Label 40.8.c.a
Level $40$
Weight $8$
Character orbit 40.c
Analytic conductor $12.495$
Analytic rank $0$
Dimension $2$
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] = [40,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.9");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(12.4954010194\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + 17 \beta q^{3} + (25 \beta + 275) q^{5} - 53 \beta q^{7} + 1031 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 17 \beta q^{3} + (25 \beta + 275) q^{5} - 53 \beta q^{7} + 1031 q^{9} - 1324 q^{11} + 4414 \beta q^{13} + (4675 \beta - 1700) q^{15} + 12000 \beta q^{17} + 4876 q^{19} + 3604 q^{21} + 23323 \beta q^{23} + (13750 \beta + 73125) q^{25} + 54706 \beta q^{27} + 110902 q^{29} - 247680 q^{31} - 22508 \beta q^{33} + ( - 14575 \beta + 5300) q^{35} - 180046 \beta q^{37} - 300152 q^{39} + 104402 q^{41} - 356811 \beta q^{43} + (25775 \beta + 283525) q^{45} - 78441 \beta q^{47} + 812307 q^{49} - 816000 q^{51} - 533134 \beta q^{53} + ( - 33100 \beta - 364100) q^{55} + 82892 \beta q^{57} - 832572 q^{59} + 529070 q^{61} - 54643 \beta q^{63} + (1213850 \beta - 441400) q^{65} + 2087209 \beta q^{67} - 1585964 q^{69} + 5176568 q^{71} + 118988 \beta q^{73} + (1243125 \beta - 935000) q^{75} + 70172 \beta q^{77} + 3742736 q^{79} - 1465211 q^{81} - 3930943 \beta q^{83} + (3300000 \beta - 1200000) q^{85} + 1885334 \beta q^{87} - 4300854 q^{89} + 935768 q^{91} - 4210560 \beta q^{93} + (121900 \beta + 1340900) q^{95} + 573896 \beta q^{97} - 1365044 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 550 q^{5} + 2062 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 550 q^{5} + 2062 q^{9} - 2648 q^{11} - 3400 q^{15} + 9752 q^{19} + 7208 q^{21} + 146250 q^{25} + 221804 q^{29} - 495360 q^{31} + 10600 q^{35} - 600304 q^{39} + 208804 q^{41} + 567050 q^{45} + 1624614 q^{49} - 1632000 q^{51} - 728200 q^{55} - 1665144 q^{59} + 1058140 q^{61} - 882800 q^{65} - 3171928 q^{69} + 10353136 q^{71} - 1870000 q^{75} + 7485472 q^{79} - 2930422 q^{81} - 2400000 q^{85} - 8601708 q^{89} + 1871536 q^{91} + 2681800 q^{95} - 2730088 q^{99}+O(q^{100}) \) 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 34.0000i 0 275.000 50.0000i 0 106.000i 0 1031.00 0
9.2 0 34.0000i 0 275.000 + 50.0000i 0 106.000i 0 1031.00 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 40.8.c.a 2
3.b odd 2 1 360.8.f.a 2
4.b odd 2 1 80.8.c.b 2
5.b even 2 1 inner 40.8.c.a 2
5.c odd 4 1 200.8.a.c 1
5.c odd 4 1 200.8.a.f 1
8.b even 2 1 320.8.c.b 2
8.d odd 2 1 320.8.c.a 2
15.d odd 2 1 360.8.f.a 2
20.d odd 2 1 80.8.c.b 2
20.e even 4 1 400.8.a.h 1
20.e even 4 1 400.8.a.n 1
40.e odd 2 1 320.8.c.a 2
40.f even 2 1 320.8.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.8.c.a 2 1.a even 1 1 trivial
40.8.c.a 2 5.b even 2 1 inner
80.8.c.b 2 4.b odd 2 1
80.8.c.b 2 20.d odd 2 1
200.8.a.c 1 5.c odd 4 1
200.8.a.f 1 5.c odd 4 1
320.8.c.a 2 8.d odd 2 1
320.8.c.a 2 40.e odd 2 1
320.8.c.b 2 8.b even 2 1
320.8.c.b 2 40.f even 2 1
360.8.f.a 2 3.b odd 2 1
360.8.f.a 2 15.d odd 2 1
400.8.a.h 1 20.e even 4 1
400.8.a.n 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1156 \) acting on \(S_{8}^{\mathrm{new}}(40, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1156 \) Copy content Toggle raw display
$5$ \( T^{2} - 550T + 78125 \) Copy content Toggle raw display
$7$ \( T^{2} + 11236 \) Copy content Toggle raw display
$11$ \( (T + 1324)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 77933584 \) Copy content Toggle raw display
$17$ \( T^{2} + 576000000 \) Copy content Toggle raw display
$19$ \( (T - 4876)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2175849316 \) Copy content Toggle raw display
$29$ \( (T - 110902)^{2} \) Copy content Toggle raw display
$31$ \( (T + 247680)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 129666248464 \) Copy content Toggle raw display
$41$ \( (T - 104402)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 509256358884 \) Copy content Toggle raw display
$47$ \( T^{2} + 24611961924 \) Copy content Toggle raw display
$53$ \( T^{2} + 1136927447824 \) Copy content Toggle raw display
$59$ \( (T + 832572)^{2} \) Copy content Toggle raw display
$61$ \( (T - 529070)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 17425765638724 \) Copy content Toggle raw display
$71$ \( (T - 5176568)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 56632576576 \) Copy content Toggle raw display
$79$ \( (T - 3742736)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 61809251476996 \) Copy content Toggle raw display
$89$ \( (T + 4300854)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1317426475264 \) Copy content Toggle raw display
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