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Properties

Label 40.6.c.a
Level $40$
Weight $6$
Character orbit 40.c
Analytic conductor $6.415$
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] = [40,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.9");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(6.41535279252\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 41x^{6} + 460x^{4} + 969x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{31}\cdot 5^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} + 1) q^{5} + (\beta_{6} + \beta_{2}) q^{7} + ( - \beta_{3} + \beta_{2} - 125) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{2} + 1) q^{5} + (\beta_{6} + \beta_{2}) q^{7} + ( - \beta_{3} + \beta_{2} - 125) q^{9} + (\beta_{4} - \beta_{2} - 92) q^{11} + (\beta_{7} + 4 \beta_{6} + \cdots - 2 \beta_1) q^{13}+ \cdots + ( - 404 \beta_{7} - 65 \beta_{4} + \cdots + 33356) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} - 1000 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} - 1000 q^{9} - 736 q^{11} - 992 q^{15} + 1376 q^{19} + 1984 q^{21} - 2136 q^{25} + 5872 q^{29} + 4224 q^{31} + 19232 q^{35} - 3008 q^{39} + 23600 q^{41} - 28328 q^{45} - 45000 q^{49} - 124800 q^{51} + 15008 q^{55} + 91680 q^{59} + 123856 q^{61} - 72064 q^{65} - 76736 q^{69} - 125632 q^{71} + 222784 q^{75} + 43264 q^{79} + 409672 q^{81} - 293760 q^{85} - 41904 q^{89} - 487616 q^{91} + 442592 q^{95} + 266848 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 41x^{6} + 460x^{4} + 969x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -182\nu^{7} - 7420\nu^{5} - 82106\nu^{3} - 160458\nu ) / 639 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 176\nu^{7} - 66\nu^{6} + 7372\nu^{5} - 2232\nu^{4} + 86042\nu^{3} - 19752\nu^{2} + 203676\nu - 29412 ) / 639 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 176\nu^{7} - 438\nu^{6} + 7372\nu^{5} - 15432\nu^{4} + 86042\nu^{3} - 113112\nu^{2} + 203676\nu + 106884 ) / 639 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 176\nu^{7} - 282\nu^{6} + 7372\nu^{5} - 17592\nu^{4} + 86042\nu^{3} - 259752\nu^{2} + 203676\nu - 390564 ) / 639 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 656\nu^{7} + 66\nu^{6} + 26548\nu^{5} + 2232\nu^{4} + 295142\nu^{3} + 19752\nu^{2} + 646692\nu + 29412 ) / 639 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 974\nu^{7} + 66\nu^{6} + 40168\nu^{5} + 2232\nu^{4} + 454172\nu^{3} + 19752\nu^{2} + 991374\nu + 29412 ) / 639 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1244 \nu^{7} - 330 \nu^{6} - 51700 \nu^{5} - 11160 \nu^{4} - 594422 \nu^{3} - 98760 \nu^{2} + \cdots - 147060 ) / 639 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 8\beta_{7} + 24\beta_{6} + 3\beta_{5} - 13\beta_{2} + 72\beta_1 ) / 1280 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{7} - \beta_{4} + 13\beta_{3} - 47\beta_{2} + 14\beta _1 - 6560 ) / 640 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 6\beta_{6} + \beta_{5} + 5\beta_{2} - 10\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 169\beta_{7} - 29\beta_{4} - 283\beta_{3} + 1157\beta_{2} - 338\beta _1 + 121760 ) / 640 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3016\beta_{7} + 10296\beta_{6} - 3443\beta_{5} - 8227\beta_{2} + 14120\beta_1 ) / 1280 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -53\beta_{7} + 16\beta_{4} + 71\beta_{3} - 352\beta_{2} + 106\beta _1 - 30496 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -57832\beta_{7} - 224376\beta_{6} + 101633\beta_{5} + 166417\beta_{2} - 282728\beta_1 ) / 1280 \) 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
1.64654i
− 0.0965878i
4.73066i
3.98753i
− 3.98753i
− 4.73066i
0.0965878i
− 1.64654i
0 − 28.9338i 0 −13.1588 − 54.3309i 0 146.828i 0 −594.165 0
9.2 0 − 24.1383i 0 46.7401 + 30.6653i 0 − 179.876i 0 −339.657 0
9.3 0 − 5.49000i 0 −53.0051 + 17.7613i 0 188.968i 0 212.860 0
9.4 0 − 4.69449i 0 23.4238 + 50.7575i 0 10.2635i 0 220.962 0
9.5 0 4.69449i 0 23.4238 − 50.7575i 0 − 10.2635i 0 220.962 0
9.6 0 5.49000i 0 −53.0051 − 17.7613i 0 − 188.968i 0 212.860 0
9.7 0 24.1383i 0 46.7401 − 30.6653i 0 179.876i 0 −339.657 0
9.8 0 28.9338i 0 −13.1588 + 54.3309i 0 − 146.828i 0 −594.165 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
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 40.6.c.a ✓ 8
3.b odd 2 1 360.6.f.b 8
4.b odd 2 1 80.6.c.d 8
5.b even 2 1 inner 40.6.c.a ✓ 8
5.c odd 4 1 200.6.a.j 4
5.c odd 4 1 200.6.a.k 4
8.b even 2 1 320.6.c.j 8
8.d odd 2 1 320.6.c.i 8
12.b even 2 1 720.6.f.n 8
15.d odd 2 1 360.6.f.b 8
20.d odd 2 1 80.6.c.d 8
20.e even 4 1 400.6.a.z 4
20.e even 4 1 400.6.a.ba 4
40.e odd 2 1 320.6.c.i 8
40.f even 2 1 320.6.c.j 8
60.h even 2 1 720.6.f.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a ✓ 8 1.a even 1 1 trivial
40.6.c.a ✓ 8 5.b even 2 1 inner
80.6.c.d 8 4.b odd 2 1
80.6.c.d 8 20.d odd 2 1
200.6.a.j 4 5.c odd 4 1
200.6.a.k 4 5.c odd 4 1
320.6.c.i 8 8.d odd 2 1
320.6.c.i 8 40.e odd 2 1
320.6.c.j 8 8.b even 2 1
320.6.c.j 8 40.f even 2 1
360.6.f.b 8 3.b odd 2 1
360.6.f.b 8 15.d odd 2 1
400.6.a.z 4 20.e even 4 1
400.6.a.ba 4 20.e even 4 1
720.6.f.n 8 12.b even 2 1
720.6.f.n 8 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 1472 T^{6} + \cdots + 324000000 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T^{4} + 368 T^{3} + \cdots + 37397137664)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 1042985883904)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 97147517576176)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 244229603328000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 296811236945008)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 11\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 51\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 64\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 93\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
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