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Properties

Label 39.5.c.a
Level $39$
Weight $5$
Character orbit 39.c
Analytic conductor $4.031$
Analytic rank $0$
Dimension $16$
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] = [39,5,Mod(14,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.14");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 39.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: \(4.03142856027\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 178 x^{14} + 12480 x^{12} + 440318 x^{10} + 8363151 x^{8} + 86725288 x^{6} + 486115240 x^{4} + \cdots + 1524412656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5}\cdot 13^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} - 6) q^{4} + (\beta_{8} - \beta_1) q^{5} + (\beta_{9} + 6) q^{6} + (\beta_{4} + \beta_{2} - 5) q^{7} + (\beta_{3} - 6 \beta_1) q^{8} + ( - \beta_{11} + \beta_{2} + \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} - 6) q^{4} + (\beta_{8} - \beta_1) q^{5} + (\beta_{9} + 6) q^{6} + (\beta_{4} + \beta_{2} - 5) q^{7} + (\beta_{3} - 6 \beta_1) q^{8} + ( - \beta_{11} + \beta_{2} + \beta_1 + 4) q^{9} + (\beta_{10} + \beta_{9} - \beta_{6} + \cdots + 15) q^{10}+ \cdots + ( - 12 \beta_{15} - 30 \beta_{14} + \cdots + 3079) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} - 100 q^{4} + 96 q^{6} - 80 q^{7} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} - 100 q^{4} + 96 q^{6} - 80 q^{7} + 66 q^{9} + 240 q^{10} - 102 q^{12} - 356 q^{15} + 456 q^{16} - 388 q^{18} - 112 q^{19} + 472 q^{21} - 316 q^{22} - 624 q^{24} - 1428 q^{25} + 1540 q^{27} + 3584 q^{28} - 1714 q^{30} - 3368 q^{31} + 1852 q^{33} - 640 q^{34} + 3862 q^{36} + 2336 q^{37} - 6776 q^{40} - 4326 q^{42} + 2956 q^{43} - 88 q^{45} + 4664 q^{46} + 1534 q^{48} + 5492 q^{49} - 8950 q^{51} - 3380 q^{52} - 14232 q^{54} + 3392 q^{55} + 4272 q^{57} + 544 q^{58} + 24444 q^{60} - 13400 q^{61} + 3452 q^{63} + 1788 q^{64} + 9504 q^{66} + 25352 q^{67} - 21296 q^{69} - 21736 q^{70} + 1908 q^{72} - 13928 q^{73} + 4540 q^{75} - 1728 q^{76} + 1690 q^{78} - 25496 q^{79} + 8346 q^{81} + 44460 q^{82} - 14384 q^{84} - 5104 q^{85} + 9332 q^{87} + 11820 q^{88} - 20634 q^{90} + 676 q^{91} - 35376 q^{93} - 30040 q^{94} + 32992 q^{96} + 39728 q^{97} + 48492 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 178 x^{14} + 12480 x^{12} + 440318 x^{10} + 8363151 x^{8} + 86725288 x^{6} + 486115240 x^{4} + \cdots + 1524412656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 38\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20352151 \nu^{14} + 3486225504 \nu^{12} + 230704736320 \nu^{10} + 7427756961426 \nu^{8} + \cdots + 49\!\cdots\!92 ) / 1585852709888 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 63329 \nu^{14} - 10951798 \nu^{12} - 735129580 \nu^{10} - 24199493086 \nu^{8} + \cdots - 20791477557824 ) / 3249698176 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2585573225 \nu^{15} - 9273317554 \nu^{14} + 450528145424 \nu^{13} - 1609421266720 \nu^{12} + \cdots - 30\!\cdots\!24 ) / 49\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2585573225 \nu^{15} + 9273317554 \nu^{14} + 450528145424 \nu^{13} + 1609421266720 \nu^{12} + \cdots + 30\!\cdots\!24 ) / 16\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 129836255 \nu^{15} + 22949389238 \nu^{13} + 1589711721720 \nu^{11} + \cdots + 67\!\cdots\!40 \nu ) / 77458993298592 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8899537 \nu^{15} - 9312753 \nu^{14} - 1544550160 \nu^{13} - 1622975600 \nu^{12} + \cdots - 38\!\cdots\!84 ) / 4757558129664 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11858890779 \nu^{15} - 106510764208 \nu^{14} + 2059949412144 \nu^{13} + \cdots - 43\!\cdots\!76 ) / 49\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 529852633 \nu^{15} + 2406167644 \nu^{14} - 92348083085 \nu^{13} + 420557263398 \nu^{12} + \cdots + 10\!\cdots\!72 ) / 154917986597184 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30405525887 \nu^{15} + 38384983432 \nu^{14} - 5278791945584 \nu^{13} + \cdots + 14\!\cdots\!72 ) / 49\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 38719945573 \nu^{15} + 77427935680 \nu^{14} + 6620464470128 \nu^{13} + \cdots + 34\!\cdots\!96 ) / 49\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25693035770 \nu^{15} + 34077309063 \nu^{14} + 4490437279744 \nu^{13} + \cdots + 15\!\cdots\!36 ) / 24\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 90419793922 \nu^{15} - 43350626617 \nu^{14} + 15662860789232 \nu^{13} + \cdots - 18\!\cdots\!48 ) / 24\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 22 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 38\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{7} + 2\beta_{6} - 3\beta_{5} - 2\beta_{4} - 55\beta_{2} + \beta _1 + 829 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} + 6 \beta_{14} + \beta_{13} - 3 \beta_{12} - 9 \beta_{11} - 17 \beta_{8} - 3 \beta_{7} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{12} - 78 \beta_{10} - 126 \beta_{9} + 66 \beta_{7} - 104 \beta_{6} + 242 \beta_{5} + \cdots - 36538 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 148 \beta_{15} - 548 \beta_{14} - 54 \beta_{13} + 270 \beta_{12} + 750 \beta_{11} + 60 \beta_{9} + \cdots + 602 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 68 \beta_{15} + 136 \beta_{14} + 136 \beta_{13} - 1820 \beta_{12} + 68 \beta_{11} + 4885 \beta_{10} + \cdots + 1697301 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 8082 \beta_{15} + 37150 \beta_{14} + 1061 \beta_{13} - 18211 \beta_{12} - 46293 \beta_{11} + \cdots - 38211 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9068 \beta_{15} - 18136 \beta_{14} - 18136 \beta_{13} + 141108 \beta_{12} - 9068 \beta_{11} + \cdots - 81020790 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 399540 \beta_{15} - 2257740 \beta_{14} + 76782 \beta_{13} + 1117790 \beta_{12} + 2580498 \beta_{11} + \cdots + 2180958 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 795032 \beta_{15} + 1590064 \beta_{14} + 1590064 \beta_{13} - 9350344 \beta_{12} + 795032 \beta_{11} + \cdots + 3939594893 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 19013818 \beta_{15} + 130095262 \beta_{14} - 11438491 \beta_{13} - 65786263 \beta_{12} + \cdots - 118656771 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 58210936 \beta_{15} - 116421872 \beta_{14} - 116421872 \beta_{13} + 571085720 \beta_{12} + \cdots - 194287209042 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 892079556 \beta_{15} - 7274294724 \beta_{14} + 975456986 \beta_{13} + 3780279174 \beta_{12} + \cdots + 6298837738 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
− 7.26283i
− 6.64731i
− 6.14277i
− 4.52706i
− 2.91748i
− 2.54095i
− 2.05736i
− 1.90680i
1.90680i
2.05736i
2.54095i
2.91748i
4.52706i
6.14277i
6.64731i
7.26283i
− 7.26283i −2.05180 + 8.76300i −36.7487 35.7440i 63.6442 + 14.9019i −65.3521 150.694i −72.5802 − 35.9599i 259.602
14.2 − 6.64731i 8.44909 − 3.10046i −28.1867 − 14.5422i −20.6097 − 56.1637i −6.77205 81.0089i 61.7743 − 52.3921i −96.6662
14.3 − 6.14277i −6.38372 − 6.34414i −21.7336 − 1.90584i −38.9706 + 39.2137i 3.68445 35.2200i 0.503752 + 80.9984i −11.7072
14.4 − 4.52706i 4.03748 + 8.04355i −4.49426 − 18.5475i 36.4136 − 18.2779i 78.3594 − 52.0872i −48.3975 + 64.9514i −83.9658
14.5 − 2.91748i −7.12425 + 5.49955i 7.48830 − 35.2981i 16.0448 + 20.7849i −84.4235 − 68.5267i 20.5099 − 78.3604i −102.982
14.6 − 2.54095i −8.58863 + 2.68988i 9.54357 32.0484i 6.83485 + 21.8233i 62.1628 − 64.9050i 66.5291 − 46.2048i 81.4334
14.7 − 2.05736i 8.97049 + 0.728222i 11.7673 39.8485i 1.49821 − 18.4555i −21.2259 − 57.1272i 79.9394 + 13.0650i 81.9825
14.8 − 1.90680i 1.69134 − 8.83965i 12.3641 − 4.03688i −16.8554 − 3.22505i −6.43312 − 54.0847i −75.2787 − 29.9017i −7.69752
14.9 1.90680i 1.69134 + 8.83965i 12.3641 4.03688i −16.8554 + 3.22505i −6.43312 54.0847i −75.2787 + 29.9017i −7.69752
14.10 2.05736i 8.97049 − 0.728222i 11.7673 − 39.8485i 1.49821 + 18.4555i −21.2259 57.1272i 79.9394 − 13.0650i 81.9825
14.11 2.54095i −8.58863 − 2.68988i 9.54357 − 32.0484i 6.83485 − 21.8233i 62.1628 64.9050i 66.5291 + 46.2048i 81.4334
14.12 2.91748i −7.12425 − 5.49955i 7.48830 35.2981i 16.0448 − 20.7849i −84.4235 68.5267i 20.5099 + 78.3604i −102.982
14.13 4.52706i 4.03748 − 8.04355i −4.49426 18.5475i 36.4136 + 18.2779i 78.3594 52.0872i −48.3975 − 64.9514i −83.9658
14.14 6.14277i −6.38372 + 6.34414i −21.7336 1.90584i −38.9706 − 39.2137i 3.68445 − 35.2200i 0.503752 − 80.9984i −11.7072
14.15 6.64731i 8.44909 + 3.10046i −28.1867 14.5422i −20.6097 + 56.1637i −6.77205 − 81.0089i 61.7743 + 52.3921i −96.6662
14.16 7.26283i −2.05180 − 8.76300i −36.7487 − 35.7440i 63.6442 − 14.9019i −65.3521 − 150.694i −72.5802 + 35.9599i 259.602
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.5.c.a ✓ 16
3.b odd 2 1 inner 39.5.c.a ✓ 16
4.b odd 2 1 624.5.f.b 16
12.b even 2 1 624.5.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.5.c.a ✓ 16 1.a even 1 1 trivial
39.5.c.a ✓ 16 3.b odd 2 1 inner
624.5.f.b 16 4.b odd 2 1
624.5.f.b 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(39, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 1524412656 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$7$ \( (T^{8} + 40 T^{7} + \cdots - 91563970928)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 42\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2197)^{8} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 21\!\cdots\!12)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 22\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 56\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 89\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 61\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 87\!\cdots\!04)^{2} \) Copy content Toggle raw display
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