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Properties

Label 40.3.g.a
Level $40$
Weight $3$
Character orbit 40.g
Analytic conductor $1.090$
Analytic rank $0$
Dimension $8$
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(11,40)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 40.g (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: \(1.08992105744\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.148996000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 7x^{6} - 2x^{5} + 12x^{3} + 47x^{2} + 114x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{6} q^{3} + ( - \beta_{5} + \beta_{4} + \beta_1 - 1) q^{4} + \beta_{5} q^{5} + (\beta_{7} - \beta_{5} - \beta_1 - 2) q^{6} + ( - \beta_{7} + \beta_{3} - \beta_1) q^{7} + (\beta_{7} + \beta_{6} - \beta_{4} + \cdots + 1) q^{8}+ \cdots + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{6} q^{3} + ( - \beta_{5} + \beta_{4} + \beta_1 - 1) q^{4} + \beta_{5} q^{5} + (\beta_{7} - \beta_{5} - \beta_1 - 2) q^{6} + ( - \beta_{7} + \beta_{3} - \beta_1) q^{7} + (\beta_{7} + \beta_{6} - \beta_{4} + \cdots + 1) q^{8}+ \cdots + (2 \beta_{6} - 6 \beta_{5} + 16 \beta_{4} + \cdots - 44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 8 q^{4} - 12 q^{6} + 8 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 8 q^{4} - 12 q^{6} + 8 q^{8} + 24 q^{9} - 10 q^{10} - 32 q^{11} + 20 q^{12} - 20 q^{14} + 8 q^{16} - 62 q^{18} + 32 q^{19} + 20 q^{20} + 20 q^{22} - 8 q^{24} - 40 q^{25} + 60 q^{26} - 96 q^{27} + 100 q^{28} + 40 q^{30} + 72 q^{32} + 16 q^{33} + 12 q^{34} - 144 q^{36} + 60 q^{38} - 40 q^{40} + 48 q^{41} + 140 q^{42} + 96 q^{43} - 88 q^{44} - 140 q^{46} - 280 q^{48} - 88 q^{49} - 10 q^{50} + 64 q^{51} - 160 q^{52} + 56 q^{54} + 280 q^{56} - 176 q^{57} - 320 q^{58} + 224 q^{59} + 100 q^{60} - 160 q^{62} - 128 q^{64} + 224 q^{66} + 160 q^{67} + 360 q^{68} + 120 q^{70} + 152 q^{72} + 160 q^{73} + 60 q^{74} - 192 q^{76} + 600 q^{78} - 160 q^{80} - 56 q^{81} + 80 q^{82} - 480 q^{83} - 40 q^{84} - 444 q^{86} - 400 q^{88} - 48 q^{89} - 210 q^{90} - 60 q^{92} - 100 q^{94} + 448 q^{96} + 224 q^{97} - 442 q^{98} - 352 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 7x^{6} - 2x^{5} + 12x^{3} + 47x^{2} + 114x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -13\nu^{7} + 31\nu^{6} + 44\nu^{5} - 250\nu^{4} + 150\nu^{3} + 234\nu^{2} + 247\nu - 2427 ) / 960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 17\nu^{6} + 44\nu^{5} - 90\nu^{4} + 150\nu^{3} - 54\nu^{2} + 71\nu + 53 ) / 160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 4\nu^{5} - 2\nu^{4} - 2\nu^{3} - 14\nu^{2} - 61\nu - 143 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 27\nu^{6} - 76\nu^{5} + 110\nu^{4} - 90\nu^{3} - 30\nu^{2} - 161\nu - 287 ) / 160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -43\nu^{7} + 193\nu^{6} - 412\nu^{5} + 410\nu^{4} - 390\nu^{3} + 54\nu^{2} - 2639\nu - 3189 ) / 960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\nu^{7} - 75\nu^{6} + 132\nu^{5} - 30\nu^{4} - 110\nu^{3} + 494\nu^{2} + 285\nu + 1399 ) / 320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\nu^{7} - 163\nu^{6} + 400\nu^{5} - 470\nu^{4} + 510\nu^{3} - 138\nu^{2} + 1379\nu + 2115 ) / 240 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{7} - \beta_{6} + 17\beta_{5} + 8\beta_{4} - 9\beta_{3} - 5\beta_{2} + 2\beta _1 - 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13\beta_{7} - 10\beta_{6} + 41\beta_{5} - 12\beta_{4} - 24\beta_{3} - 24\beta_{2} - \beta _1 - 58 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} - 20\beta_{6} + 35\beta_{5} - 49\beta_{4} - 36\beta_{3} - 26\beta_{2} - 15\beta _1 - 92 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -58\beta_{7} - 92\beta_{6} + 47\beta_{5} - 345\beta_{4} - 144\beta_{3} - 68\beta_{2} - 183\beta _1 - 633 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
1.01379 + 1.99995i
1.01379 − 1.99995i
−0.981166 + 0.273826i
−0.981166 − 0.273826i
−0.528640 − 1.28967i
−0.528640 + 1.28967i
2.49601 − 1.32738i
2.49601 + 1.32738i
−1.41908 − 1.40933i 4.13348 0.0275898 + 3.99990i − 2.23607i −5.86575 − 5.82543i − 2.28473i 5.59803 − 5.71508i 8.08569 −3.15135 + 3.17316i
11.2 −1.41908 + 1.40933i 4.13348 0.0275898 − 3.99990i 2.23607i −5.86575 + 5.82543i 2.28473i 5.59803 + 5.71508i 8.08569 −3.15135 − 3.17316i
11.3 −0.137237 − 1.99529i −5.01222 −3.96233 + 0.547652i − 2.23607i 0.687859 + 10.0008i − 8.92613i 1.63650 + 7.83083i 16.1223 −4.46160 + 0.306870i
11.4 −0.137237 + 1.99529i −5.01222 −3.96233 − 0.547652i 2.23607i 0.687859 − 10.0008i 8.92613i 1.63650 − 7.83083i 16.1223 −4.46160 − 0.306870i
11.5 0.686557 − 1.87847i 2.08343 −3.05728 − 2.57935i 2.23607i 1.43039 − 3.91366i 1.18656i −6.94422 + 3.97213i −4.65931 4.20038 + 1.53519i
11.6 0.686557 + 1.87847i 2.08343 −3.05728 + 2.57935i − 2.23607i 1.43039 + 3.91366i − 1.18656i −6.94422 − 3.97213i −4.65931 4.20038 − 1.53519i
11.7 1.86976 − 0.709922i −1.20470 2.99202 − 2.65477i − 2.23607i −2.25250 + 0.855241i 12.3974i 3.70969 − 7.08789i −7.54870 −1.58743 − 4.18092i
11.8 1.86976 + 0.709922i −1.20470 2.99202 + 2.65477i 2.23607i −2.25250 − 0.855241i − 12.3974i 3.70969 + 7.08789i −7.54870 −1.58743 + 4.18092i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.3.g.a ✓ 8
3.b odd 2 1 360.3.g.a 8
4.b odd 2 1 160.3.g.a 8
5.b even 2 1 200.3.g.g 8
5.c odd 4 2 200.3.e.d 16
8.b even 2 1 160.3.g.a 8
8.d odd 2 1 inner 40.3.g.a ✓ 8
12.b even 2 1 1440.3.g.a 8
16.e even 4 2 1280.3.b.i 16
16.f odd 4 2 1280.3.b.i 16
20.d odd 2 1 800.3.g.g 8
20.e even 4 2 800.3.e.d 16
24.f even 2 1 360.3.g.a 8
24.h odd 2 1 1440.3.g.a 8
40.e odd 2 1 200.3.g.g 8
40.f even 2 1 800.3.g.g 8
40.i odd 4 2 800.3.e.d 16
40.k even 4 2 200.3.e.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.g.a ✓ 8 1.a even 1 1 trivial
40.3.g.a ✓ 8 8.d odd 2 1 inner
160.3.g.a 8 4.b odd 2 1
160.3.g.a 8 8.b even 2 1
200.3.e.d 16 5.c odd 4 2
200.3.e.d 16 40.k even 4 2
200.3.g.g 8 5.b even 2 1
200.3.g.g 8 40.e odd 2 1
360.3.g.a 8 3.b odd 2 1
360.3.g.a 8 24.f even 2 1
800.3.e.d 16 20.e even 4 2
800.3.e.d 16 40.i odd 4 2
800.3.g.g 8 20.d odd 2 1
800.3.g.g 8 40.f even 2 1
1280.3.b.i 16 16.e even 4 2
1280.3.b.i 16 16.f odd 4 2
1440.3.g.a 8 12.b even 2 1
1440.3.g.a 8 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} - 24 T^{2} + \cdots + 52)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 240 T^{6} + \cdots + 90000 \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{3} + \cdots - 7792)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 880 T^{6} + \cdots + 138297600 \) Copy content Toggle raw display
$17$ \( (T^{4} - 664 T^{2} + \cdots + 72592)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 16 T^{3} + \cdots - 2672)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2320 T^{6} + \cdots + 47610000 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 184968806400 \) Copy content Toggle raw display
$31$ \( T^{8} + 2240 T^{6} + \cdots + 23040000 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 134307590400 \) Copy content Toggle raw display
$41$ \( (T^{4} - 24 T^{3} + \cdots - 1472)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 48 T^{3} + \cdots + 5598004)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 1101366291600 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 3957553209600 \) Copy content Toggle raw display
$59$ \( (T^{4} - 112 T^{3} + \cdots - 4307824)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 39126526214400 \) Copy content Toggle raw display
$67$ \( (T^{4} - 80 T^{3} + \cdots - 22028)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 2536502169600 \) Copy content Toggle raw display
$73$ \( (T^{4} - 80 T^{3} + \cdots - 12225008)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 976317515366400 \) Copy content Toggle raw display
$83$ \( (T^{4} + 240 T^{3} + \cdots - 721708)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{3} + \cdots - 3224432)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 112 T^{3} + \cdots + 83984)^{2} \) Copy content Toggle raw display
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