Properties

Label 40.10.a.d
Level $40$
Weight $10$
Character orbit 40.a
Self dual yes
Analytic conductor $20.601$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [40,10,Mod(1,40)]
 
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, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 40.a (trivial)

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: yes
Analytic conductor: \(20.6014334466\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.7117.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 19x - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{3}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 28) q^{3} + 625 q^{5} + ( - \beta_{2} + 17 \beta_1 - 1840) q^{7} + (4 \beta_{2} + 76 \beta_1 + 15693) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 28) q^{3} + 625 q^{5} + ( - \beta_{2} + 17 \beta_1 - 1840) q^{7} + (4 \beta_{2} + 76 \beta_1 + 15693) q^{9} + (\beta_{2} + 292 \beta_1 + 1852) q^{11} + ( - 16 \beta_{2} - 28 \beta_1 - 27698) q^{13} + (625 \beta_1 + 17500) q^{15} + ( - 1596 \beta_1 + 122354) q^{17} + (81 \beta_{2} - 378 \beta_1 + 496372) q^{19} + (60 \beta_{2} - 5248 \beta_1 + 524576) q^{21} + ( - 319 \beta_{2} - 1417 \beta_1 - 166640) q^{23} + 390625 q^{25} + (336 \beta_{2} + 16554 \beta_1 + 2565144) q^{27} + ( - 380 \beta_{2} - 6128 \beta_1 + 1744894) q^{29} + (238 \beta_{2} + 6022 \beta_1 + 4236304) q^{31} + (1176 \beta_{2} + 20092 \beta_1 + 10164688) q^{33} + ( - 625 \beta_{2} + 10625 \beta_1 - 1150000) q^{35} + ( - 1544 \beta_{2} + 50680 \beta_1 + 7241478) q^{37} + ( - 240 \beta_{2} - 96626 \beta_1 - 1935608) q^{39} + ( - 556 \beta_{2} + 3188 \beta_1 + 9146826) q^{41} + (1134 \beta_{2} - 162615 \beta_1 - 7739420) q^{43} + (2500 \beta_{2} + 47500 \beta_1 + 9808125) q^{45} + (2081 \beta_{2} + 115901 \beta_1 - 9567176) q^{47} + (1292 \beta_{2} - 218524 \beta_1 + 9031641) q^{49} + ( - 6384 \beta_{2} + 45746 \beta_1 - 51782920) q^{51} + ( - 5104 \beta_{2} - 175444 \beta_1 - 15209994) q^{53} + (625 \beta_{2} + 182500 \beta_1 + 1157500) q^{55} + ( - 864 \beta_{2} + 820372 \beta_1 + 1792048) q^{57} + (5775 \beta_{2} + 183246 \beta_1 - 89573956) q^{59} + (12800 \beta_{2} - 362224 \beta_1 + 51990046) q^{61} + ( - 829 \beta_{2} + 191501 \beta_1 - 129915888) q^{63} + ( - 10000 \beta_{2} - 17500 \beta_1 - 17311250) q^{65} + ( - 4270 \beta_{2} + 277043 \beta_1 - 175638868) q^{67} + ( - 8220 \beta_{2} - 1582112 \beta_1 - 57500576) q^{69} + (12320 \beta_{2} - 772366 \beta_1 - 79964808) q^{71} + ( - 4832 \beta_{2} + 460180 \beta_1 + 66120810) q^{73} + (390625 \beta_1 + 10937500) q^{75} + (25888 \beta_{2} - 1549988 \beta_1 + 128606336) q^{77} + ( - 29392 \beta_{2} + 298676 \beta_1 - 137946576) q^{79} + ( - 9828 \beta_{2} + 3283092 \beta_1 + 339595929) q^{81} + (21868 \beta_{2} - 2169743 \beta_1 + 123983276) q^{83} + ( - 997500 \beta_1 + 76471250) q^{85} + ( - 27552 \beta_{2} - 154370 \beta_1 - 167670584) q^{87} + (20184 \beta_{2} + 2366280 \beta_1 + 251642202) q^{89} + ( - 7470 \beta_{2} - 1749570 \beta_1 + 614129888) q^{91} + (25992 \beta_{2} + 5530672 \beta_1 + 329777920) q^{93} + (50625 \beta_{2} - 236250 \beta_1 + 310232500) q^{95} + ( - 18648 \beta_{2} - 1338708 \beta_1 - 301150334) q^{97} + (70093 \beta_{2} + 10349092 \beta_1 + 957255180) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 84 q^{3} + 1875 q^{5} - 5520 q^{7} + 47079 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 84 q^{3} + 1875 q^{5} - 5520 q^{7} + 47079 q^{9} + 5556 q^{11} - 83094 q^{13} + 52500 q^{15} + 367062 q^{17} + 1489116 q^{19} + 1573728 q^{21} - 499920 q^{23} + 1171875 q^{25} + 7695432 q^{27} + 5234682 q^{29} + 12708912 q^{31} + 30494064 q^{33} - 3450000 q^{35} + 21724434 q^{37} - 5806824 q^{39} + 27440478 q^{41} - 23218260 q^{43} + 29424375 q^{45} - 28701528 q^{47} + 27094923 q^{49} - 155348760 q^{51} - 45629982 q^{53} + 3472500 q^{55} + 5376144 q^{57} - 268721868 q^{59} + 155970138 q^{61} - 389747664 q^{63} - 51933750 q^{65} - 526916604 q^{67} - 172501728 q^{69} - 239894424 q^{71} + 198362430 q^{73} + 32812500 q^{75} + 385819008 q^{77} - 413839728 q^{79} + 1018787787 q^{81} + 371949828 q^{83} + 229413750 q^{85} - 503011752 q^{87} + 754926606 q^{89} + 1842389664 q^{91} + 989333760 q^{93} + 930697500 q^{95} - 903451002 q^{97} + 2871765540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 19x - 22 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -12\nu^{2} + 84\nu + 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 1272\nu^{2} - 3144\nu - 15488 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 106\beta _1 + 1920 ) / 5760 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{2} + 262\beta _1 + 74880 ) / 5760 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.92453
−1.41013
5.33466
0 −192.296 0 625.000 0 −10171.1 0 17294.6 0
1.2 0 13.6876 0 625.000 0 6441.91 0 −19495.7 0
1.3 0 262.608 0 625.000 0 −1790.86 0 49280.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.10.a.d 3
3.b odd 2 1 360.10.a.j 3
4.b odd 2 1 80.10.a.k 3
5.b even 2 1 200.10.a.f 3
5.c odd 4 2 200.10.c.f 6
8.b even 2 1 320.10.a.u 3
8.d odd 2 1 320.10.a.v 3
20.d odd 2 1 400.10.a.x 3
20.e even 4 2 400.10.c.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.10.a.d 3 1.a even 1 1 trivial
80.10.a.k 3 4.b odd 2 1
200.10.a.f 3 5.b even 2 1
200.10.c.f 6 5.c odd 4 2
320.10.a.u 3 8.b even 2 1
320.10.a.v 3 8.d odd 2 1
360.10.a.j 3 3.b odd 2 1
400.10.a.x 3 20.d odd 2 1
400.10.c.r 6 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 84T_{3}^{2} - 49536T_{3} + 691200 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(40))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 84 T^{2} - 49536 T + 691200 \) Copy content Toggle raw display
$5$ \( (T - 625)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 5520 T^{2} + \cdots - 117339254144 \) Copy content Toggle raw display
$11$ \( T^{3} - 5556 T^{2} + \cdots - 46675499449024 \) Copy content Toggle raw display
$13$ \( T^{3} + 83094 T^{2} + \cdots - 10\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{3} - 367062 T^{2} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{3} - 1489116 T^{2} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{3} + 499920 T^{2} + \cdots - 59\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{3} - 5234682 T^{2} + \cdots + 81\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{3} - 12708912 T^{2} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} - 21724434 T^{2} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{3} - 27440478 T^{2} + \cdots - 62\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{3} + 23218260 T^{2} + \cdots - 20\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{3} + 28701528 T^{2} + \cdots - 19\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{3} + 45629982 T^{2} + \cdots + 76\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{3} + 268721868 T^{2} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{3} - 155970138 T^{2} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{3} + 526916604 T^{2} + \cdots + 46\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{3} + 239894424 T^{2} + \cdots - 54\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{3} - 198362430 T^{2} + \cdots + 89\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{3} + 413839728 T^{2} + \cdots - 67\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} - 371949828 T^{2} + \cdots - 65\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{3} - 754926606 T^{2} + \cdots + 71\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{3} + 903451002 T^{2} + \cdots + 77\!\cdots\!04 \) Copy content Toggle raw display
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