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Properties

Label 39.6.j.b
Level $39$
Weight $6$
Character orbit 39.j
Analytic conductor $6.255$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [39,6,Mod(4,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("39.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 39.j (of order \(6\), 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: \(6.25496897271\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 310x^{10} + 35389x^{8} + 1834428x^{6} + 43601364x^{4} + 433521504x^{2} + 1332542016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + 9 \beta_{3} q^{3} + ( - \beta_{5} - 20 \beta_{3} + 20) q^{4} + ( - \beta_{7} - \beta_{6} + 20 \beta_{3} - 10) q^{5} - 9 \beta_{2} q^{6} + ( - \beta_{11} + \beta_{5} + \beta_{4} + \cdots + 23) q^{7}+ \cdots + (81 \beta_{11} + 81 \beta_{10} + \cdots - 1782) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 54 q^{3} + 118 q^{4} + 414 q^{7} - 486 q^{9} - 56 q^{10} + 360 q^{11} + 2124 q^{12} + 986 q^{13} - 3204 q^{14} - 1620 q^{15} - 1706 q^{16} + 528 q^{17} - 5880 q^{19} + 6678 q^{20} + 322 q^{22} + 6216 q^{23}+ \cdots - 402804 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 310x^{10} + 35389x^{8} + 1834428x^{6} + 43601364x^{4} + 433521504x^{2} + 1332542016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 155\nu^{4} + 5682\nu^{2} + 2496\nu + 36504 ) / 4992 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 310\nu^{9} + 35389\nu^{7} + 1797924\nu^{5} + 37943244\nu^{3} + 226105776\nu + 91113984 ) / 182227968 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{2} - 52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} - 310 \nu^{9} - 34687 \nu^{7} - 1689114 \nu^{5} - 33954480 \nu^{3} + 1752192 \nu^{2} + \cdots + 91113984 ) / 3504384 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 181 \nu^{11} + 2028 \nu^{10} + 54082 \nu^{9} + 555672 \nu^{8} + 5715889 \nu^{7} + \cdots + 93107102400 ) / 668169216 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 181 \nu^{11} - 2028 \nu^{10} + 54082 \nu^{9} - 555672 \nu^{8} + 5715889 \nu^{7} + \cdots - 93107102400 ) / 668169216 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 604 \nu^{11} + 1521 \nu^{10} + 172030 \nu^{9} + 416754 \nu^{8} + 17408188 \nu^{7} + \cdots + 345324846672 ) / 1002253824 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 604 \nu^{11} + 1521 \nu^{10} - 172030 \nu^{9} + 416754 \nu^{8} - 17408188 \nu^{7} + \cdots + 346327100496 ) / 1002253824 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1138 \nu^{11} - 7605 \nu^{10} - 328444 \nu^{9} - 2284542 \nu^{8} - 33403846 \nu^{7} + \cdots - 796094198640 ) / 1002253824 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1138 \nu^{11} + 7605 \nu^{10} - 328444 \nu^{9} + 2284542 \nu^{8} - 33403846 \nu^{7} + \cdots + 796094198640 ) / 1002253824 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} - 52 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 3\beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} + 2\beta_{3} - 81\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{9} + 4\beta_{8} + 2\beta_{7} - 2\beta_{6} + 113\beta_{4} - 80\beta_{2} + 40\beta _1 + 4258 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 119 \beta_{11} - 119 \beta_{10} + 107 \beta_{9} - 107 \beta_{8} - 417 \beta_{7} - 417 \beta_{6} + \cdots + 2112 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 620 \beta_{9} - 620 \beta_{8} - 310 \beta_{7} + 310 \beta_{6} - 11833 \beta_{4} + 17392 \beta_{2} + \cdots - 401030 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12763 \beta_{11} + 12763 \beta_{10} - 10903 \beta_{9} + 10903 \beta_{8} + 47589 \beta_{7} + \cdots - 457152 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2496 \beta_{11} - 2496 \beta_{10} + 75868 \beta_{9} + 75868 \beta_{8} + 44174 \beta_{7} + \cdots + 39868894 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1338611 \beta_{11} - 1338611 \beta_{10} + 1148447 \beta_{9} - 1148447 \beta_{8} - 5131389 \beta_{7} + \cdots + 69466176 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 683904 \beta_{11} + 683904 \beta_{10} - 8649788 \beta_{9} - 8649788 \beta_{8} - 6199390 \beta_{7} + \cdots - 4072808942 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 139309315 \beta_{11} + 139309315 \beta_{10} - 124606927 \beta_{9} + 124606927 \beta_{8} + \cdots - 9244691904 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(1 - \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 9.68706i
− 7.60446i
− 2.32967i
3.82029i
5.30461i
10.4963i
9.68706i
7.60446i
2.32967i
− 3.82029i
− 5.30461i
− 10.4963i
−8.38924 − 4.84353i 4.50000 − 7.79423i 30.9196 + 53.5543i − 92.4599i −75.5032 + 43.5918i 209.382 − 120.887i − 289.054i −40.5000 − 70.1481i −447.832 + 775.669i
4.2 −6.58566 − 3.80223i 4.50000 − 7.79423i 12.9139 + 22.3675i 60.8553i −59.2709 + 34.2201i −28.1411 + 16.2473i 46.9362i −40.5000 − 70.1481i 231.386 − 400.772i
4.3 −2.01755 − 1.16483i 4.50000 − 7.79423i −13.2863 − 23.0126i − 3.62321i −18.1580 + 10.4835i −67.5002 + 38.9713i 136.455i −40.5000 − 70.1481i −4.22043 + 7.31001i
4.4 3.30847 + 1.91015i 4.50000 − 7.79423i −8.70269 − 15.0735i 26.1960i 29.7762 − 17.1913i 210.299 − 121.416i − 188.743i −40.5000 − 70.1481i −50.0383 + 86.6688i
4.5 4.59393 + 2.65231i 4.50000 − 7.79423i −1.93054 − 3.34380i − 98.3488i 41.3454 − 23.8708i −133.821 + 77.2615i − 190.229i −40.5000 − 70.1481i 260.851 − 451.808i
4.6 9.09005 + 5.24814i 4.50000 − 7.79423i 39.0860 + 67.6990i 3.45756i 81.8105 − 47.2333i 16.7808 − 9.68841i 484.636i −40.5000 − 70.1481i −18.1458 + 31.4294i
10.1 −8.38924 + 4.84353i 4.50000 + 7.79423i 30.9196 − 53.5543i 92.4599i −75.5032 − 43.5918i 209.382 + 120.887i 289.054i −40.5000 + 70.1481i −447.832 − 775.669i
10.2 −6.58566 + 3.80223i 4.50000 + 7.79423i 12.9139 − 22.3675i − 60.8553i −59.2709 − 34.2201i −28.1411 − 16.2473i − 46.9362i −40.5000 + 70.1481i 231.386 + 400.772i
10.3 −2.01755 + 1.16483i 4.50000 + 7.79423i −13.2863 + 23.0126i 3.62321i −18.1580 − 10.4835i −67.5002 − 38.9713i − 136.455i −40.5000 + 70.1481i −4.22043 − 7.31001i
10.4 3.30847 − 1.91015i 4.50000 + 7.79423i −8.70269 + 15.0735i − 26.1960i 29.7762 + 17.1913i 210.299 + 121.416i 188.743i −40.5000 + 70.1481i −50.0383 − 86.6688i
10.5 4.59393 − 2.65231i 4.50000 + 7.79423i −1.93054 + 3.34380i 98.3488i 41.3454 + 23.8708i −133.821 − 77.2615i 190.229i −40.5000 + 70.1481i 260.851 + 451.808i
10.6 9.09005 − 5.24814i 4.50000 + 7.79423i 39.0860 − 67.6990i − 3.45756i 81.8105 + 47.2333i 16.7808 + 9.68841i − 484.636i −40.5000 + 70.1481i −18.1458 − 31.4294i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.6.j.b ✓ 12
3.b odd 2 1 117.6.q.d 12
13.e even 6 1 inner 39.6.j.b ✓ 12
13.f odd 12 2 507.6.a.m 12
39.h odd 6 1 117.6.q.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.6.j.b ✓ 12 1.a even 1 1 trivial
39.6.j.b ✓ 12 13.e even 6 1 inner
117.6.q.d 12 3.b odd 2 1
117.6.q.d 12 39.h odd 6 1
507.6.a.m 12 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 155 T_{2}^{10} + 18343 T_{2}^{8} + 7488 T_{2}^{7} - 807702 T_{2}^{6} + 26627004 T_{2}^{4} + \cdots + 1332542016 \) acting on \(S_{6}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 1332542016 \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 47\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 75\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 96\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 31\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 12\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 42\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 81\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
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