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Properties

Label 40.4.c.a
Level $40$
Weight $4$
Character orbit 40.c
Analytic conductor $2.360$
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] = [40,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(2.36007640023\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{9} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 20) q^{11} + (7 \beta_{2} + 5 \beta_1) q^{13} + ( - 5 \beta_{2} - 10 \beta_1 - 20) q^{15} + ( - 2 \beta_{2} - 18 \beta_1) q^{17} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 20) q^{19} + ( - 2 \beta_{3} - \beta_{2} + \cdots - 68) q^{21}+ \cdots + ( - 44 \beta_{3} - 22 \beta_{2} + \cdots - 788) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 4 q^{9} + 80 q^{11} - 80 q^{15} - 80 q^{19} - 272 q^{21} + 276 q^{25} + 280 q^{29} + 384 q^{31} - 304 q^{35} - 224 q^{39} - 1048 q^{41} + 772 q^{45} + 492 q^{49} + 1920 q^{51} - 1616 q^{55} - 1392 q^{59} - 1384 q^{61} + 608 q^{65} + 112 q^{69} + 1376 q^{71} + 160 q^{75} + 1472 q^{79} - 1484 q^{81} + 1152 q^{85} + 1320 q^{89} + 992 q^{91} - 1456 q^{95} - 3152 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 2\nu^{2} + 6\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 6\nu^{2} - 6\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} - 4\nu^{2} + 12\nu ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 3\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} - 3\beta_{2} + 3\beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

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

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

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
1.22474 − 1.22474i
−1.22474 − 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
0 − 6.89898i 0 −10.7980 − 2.89898i 0 − 12.6969i 0 −20.5959 0
9.2 0 − 2.89898i 0 8.79796 − 6.89898i 0 − 16.6969i 0 18.5959 0
9.3 0 2.89898i 0 8.79796 + 6.89898i 0 16.6969i 0 18.5959 0
9.4 0 6.89898i 0 −10.7980 + 2.89898i 0 12.6969i 0 −20.5959 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 40.4.c.a ✓ 4
3.b odd 2 1 360.4.f.e 4
4.b odd 2 1 80.4.c.c 4
5.b even 2 1 inner 40.4.c.a ✓ 4
5.c odd 4 1 200.4.a.k 2
5.c odd 4 1 200.4.a.l 2
8.b even 2 1 320.4.c.g 4
8.d odd 2 1 320.4.c.h 4
12.b even 2 1 720.4.f.m 4
15.d odd 2 1 360.4.f.e 4
15.e even 4 1 1800.4.a.bk 2
15.e even 4 1 1800.4.a.bp 2
20.d odd 2 1 80.4.c.c 4
20.e even 4 1 400.4.a.v 2
20.e even 4 1 400.4.a.x 2
40.e odd 2 1 320.4.c.h 4
40.f even 2 1 320.4.c.g 4
40.i odd 4 1 1600.4.a.ce 2
40.i odd 4 1 1600.4.a.cl 2
40.k even 4 1 1600.4.a.cf 2
40.k even 4 1 1600.4.a.cm 2
60.h even 2 1 720.4.f.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.c.a ✓ 4 1.a even 1 1 trivial
40.4.c.a ✓ 4 5.b even 2 1 inner
80.4.c.c 4 4.b odd 2 1
80.4.c.c 4 20.d odd 2 1
200.4.a.k 2 5.c odd 4 1
200.4.a.l 2 5.c odd 4 1
320.4.c.g 4 8.b even 2 1
320.4.c.g 4 40.f even 2 1
320.4.c.h 4 8.d odd 2 1
320.4.c.h 4 40.e odd 2 1
360.4.f.e 4 3.b odd 2 1
360.4.f.e 4 15.d odd 2 1
400.4.a.v 2 20.e even 4 1
400.4.a.x 2 20.e even 4 1
720.4.f.m 4 12.b even 2 1
720.4.f.m 4 60.h even 2 1
1600.4.a.ce 2 40.i odd 4 1
1600.4.a.cf 2 40.k even 4 1
1600.4.a.cl 2 40.i odd 4 1
1600.4.a.cm 2 40.k even 4 1
1800.4.a.bk 2 15.e even 4 1
1800.4.a.bp 2 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{4} + 440 T^{2} + 44944 \) Copy content Toggle raw display
$11$ \( (T^{2} - 40 T - 1136)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 5600 T^{2} + 6801664 \) Copy content Toggle raw display
$17$ \( T^{4} + 16896 T^{2} + 14745600 \) Copy content Toggle raw display
$19$ \( (T^{2} + 40 T - 1136)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 48440 T^{2} + 259467664 \) Copy content Toggle raw display
$29$ \( (T^{2} - 140 T - 1244)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 192 T - 46080)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 75488 T^{2} + 314849536 \) Copy content Toggle raw display
$41$ \( (T^{2} + 524 T + 65188)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 90360 T^{2} + 576576144 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 9927332496 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 72090102016 \) Copy content Toggle raw display
$59$ \( (T^{2} + 696 T - 64752)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 692 T + 100900)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 52280 T^{2} + 565868944 \) Copy content Toggle raw display
$71$ \( (T^{2} - 688 T + 93760)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 9130184704 \) Copy content Toggle raw display
$79$ \( (T^{2} - 736 T + 129280)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 365991120784 \) Copy content Toggle raw display
$89$ \( (T^{2} - 660 T + 53604)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 26342588416 \) Copy content Toggle raw display
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