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Properties

Label 40.7.l.b
Level $40$
Weight $7$
Character orbit 40.l
Analytic conductor $9.202$
Analytic rank $0$
Dimension $10$
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] = [40,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.17");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(9.20216334479\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5886 x^{7} + 239880 x^{6} - 2872564 x^{5} + 17322498 x^{4} - 9255644 x^{3} + 338486404 x^{2} + \cdots + 24969867392 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{19}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 + 1) q^{3} + ( - \beta_{3} + \beta_{2} - 16 \beta_1 + 9) q^{5} + ( - \beta_{8} + \beta_{5} + 2 \beta_{4} + \cdots + 45) q^{7}+ \cdots + (\beta_{8} + \beta_{7} + \cdots + 247 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1 + 1) q^{3} + ( - \beta_{3} + \beta_{2} - 16 \beta_1 + 9) q^{5} + ( - \beta_{8} + \beta_{5} + 2 \beta_{4} + \cdots + 45) q^{7}+ \cdots + ( - 747 \beta_{9} - 98 \beta_{8} + \cdots - 1494) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 14 q^{3} + 94 q^{5} + 454 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 14 q^{3} + 94 q^{5} + 454 q^{7} - 2708 q^{11} + 2282 q^{13} + 4918 q^{15} - 6878 q^{17} + 21564 q^{21} - 42 q^{23} + 23570 q^{25} - 35536 q^{27} - 110932 q^{31} + 25484 q^{33} + 112638 q^{35} - 60478 q^{37} - 219412 q^{41} + 101854 q^{43} + 595524 q^{45} - 246170 q^{47} - 539460 q^{51} + 136290 q^{53} + 570972 q^{55} - 1072832 q^{57} - 741348 q^{61} + 1632390 q^{63} + 2042954 q^{65} - 676786 q^{67} - 2695732 q^{71} + 416378 q^{73} + 2910734 q^{75} - 2645028 q^{77} - 2353310 q^{81} + 1992446 q^{83} + 4727322 q^{85} - 1595264 q^{87} - 3378500 q^{91} - 2346340 q^{93} + 856336 q^{95} - 3187830 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5886 x^{7} + 239880 x^{6} - 2872564 x^{5} + 17322498 x^{4} - 9255644 x^{3} + 338486404 x^{2} + \cdots + 24969867392 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17\!\cdots\!99 \nu^{9} + \cdots + 35\!\cdots\!92 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 92160334667705 \nu^{9} + \cdots - 21\!\cdots\!06 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 59\!\cdots\!69 \nu^{9} + \cdots + 23\!\cdots\!04 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!81 \nu^{9} + \cdots + 10\!\cdots\!72 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!45 \nu^{9} + \cdots + 27\!\cdots\!32 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 58\!\cdots\!91 \nu^{9} + \cdots + 53\!\cdots\!16 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 61\!\cdots\!71 \nu^{9} + \cdots - 26\!\cdots\!84 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 65\!\cdots\!89 \nu^{9} + \cdots + 65\!\cdots\!56 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 82\!\cdots\!15 \nu^{9} + \cdots - 28\!\cdots\!28 ) / 27\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{6} - 4\beta_{5} + 5\beta_{4} + \beta_{3} + 4\beta_{2} + 2\beta _1 - 2 ) / 80 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{9} - 10 \beta_{8} - 10 \beta_{7} - 7 \beta_{6} - 8 \beta_{5} - 43 \beta_{4} - 35 \beta_{3} + \cdots + 6 ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 77 \beta_{9} + 110 \beta_{7} + 138 \beta_{6} + 613 \beta_{5} + 45 \beta_{4} + 568 \beta_{3} + \cdots + 35514 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1216 \beta_{9} + 1123 \beta_{8} - 1123 \beta_{7} - 439 \beta_{6} - 5987 \beta_{5} + 4317 \beta_{4} + \cdots - 772184 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 165407 \beta_{9} - 174580 \beta_{8} - 83577 \beta_{6} + 427058 \beta_{5} - 781095 \beta_{4} + \cdots + 57899014 ) / 40 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 729211 \beta_{9} + 1708485 \beta_{8} + 1708485 \beta_{7} + 2111524 \beta_{6} + 2666781 \beta_{5} + \cdots - 1458422 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 52114821 \beta_{9} - 118596690 \beta_{7} - 105162199 \beta_{6} - 473696174 \beta_{5} + \cdots - 39467845122 ) / 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 279417684 \beta_{9} - 218322847 \beta_{8} + 218322847 \beta_{7} + 95394026 \beta_{6} + \cdots + 145995572312 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 34034054749 \beta_{9} + 39013713120 \beta_{8} + 16788473079 \beta_{6} - 88922023766 \beta_{5} + \cdots - 13039111768298 ) / 20 \) 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
9.12538 + 9.12538i
−18.0462 − 18.0462i
−4.35031 − 4.35031i
4.39091 + 4.39091i
8.88019 + 8.88019i
9.12538 − 9.12538i
−18.0462 + 18.0462i
−4.35031 + 4.35031i
4.39091 − 4.39091i
8.88019 − 8.88019i
0 −30.7007 + 30.7007i 0 −32.4901 + 120.704i 0 70.1144 + 70.1144i 0 − 1156.06i 0
17.2 0 −8.22680 + 8.22680i 0 −37.8160 − 119.143i 0 327.073 + 327.073i 0 593.640i 0
17.3 0 −4.14542 + 4.14542i 0 124.252 − 13.6571i 0 −263.094 − 263.094i 0 694.631i 0
17.4 0 16.0376 − 16.0376i 0 −119.677 + 36.0896i 0 −332.355 − 332.355i 0 214.589i 0
17.5 0 34.0353 − 34.0353i 0 112.731 + 54.0063i 0 425.262 + 425.262i 0 − 1587.80i 0
33.1 0 −30.7007 − 30.7007i 0 −32.4901 − 120.704i 0 70.1144 − 70.1144i 0 1156.06i 0
33.2 0 −8.22680 − 8.22680i 0 −37.8160 + 119.143i 0 327.073 − 327.073i 0 − 593.640i 0
33.3 0 −4.14542 − 4.14542i 0 124.252 + 13.6571i 0 −263.094 + 263.094i 0 − 694.631i 0
33.4 0 16.0376 + 16.0376i 0 −119.677 − 36.0896i 0 −332.355 + 332.355i 0 − 214.589i 0
33.5 0 34.0353 + 34.0353i 0 112.731 − 54.0063i 0 425.262 − 425.262i 0 1587.80i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.5
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 40.7.l.b ✓ 10
3.b odd 2 1 360.7.v.b 10
4.b odd 2 1 80.7.p.f 10
5.b even 2 1 200.7.l.f 10
5.c odd 4 1 inner 40.7.l.b ✓ 10
5.c odd 4 1 200.7.l.f 10
15.e even 4 1 360.7.v.b 10
20.e even 4 1 80.7.p.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.7.l.b ✓ 10 1.a even 1 1 trivial
40.7.l.b ✓ 10 5.c odd 4 1 inner
80.7.p.f 10 4.b odd 2 1
80.7.p.f 10 20.e even 4 1
200.7.l.f 10 5.b even 2 1
200.7.l.f 10 5.c odd 4 1
360.7.v.b 10 3.b odd 2 1
360.7.v.b 10 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 14 T_{3}^{9} + 98 T_{3}^{8} + 18192 T_{3}^{7} + 4343076 T_{3}^{6} - 31540536 T_{3}^{5} + \cdots + 10451592000000 \) acting on \(S_{7}^{\mathrm{new}}(40, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 10451592000000 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 23\!\cdots\!88 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 600218727318016)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 24\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 11\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 82\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 20\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 76\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 37\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 62\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 69\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 68\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 21\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 53\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 82\!\cdots\!72 \) Copy content Toggle raw display
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