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Properties

Label 39.5.g.a
Level $39$
Weight $5$
Character orbit 39.g
Analytic conductor $4.031$
Analytic rank $0$
Dimension $20$
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(31,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.31");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 39.g (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: \(4.03142856027\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{15} + \beta_{9} + 10 \beta_{4}) q^{4} + ( - \beta_{13} + \beta_{4} + 1) q^{5} + (\beta_{5} - \beta_1) q^{6} + ( - \beta_{12} - 2 \beta_{9} + \cdots - 1) q^{7}+ \cdots + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{15} + \beta_{9} + 10 \beta_{4}) q^{4} + ( - \beta_{13} + \beta_{4} + 1) q^{5} + (\beta_{5} - \beta_1) q^{6} + ( - \beta_{12} - 2 \beta_{9} + \cdots - 1) q^{7}+ \cdots + ( - 27 \beta_{16} - 81 \beta_{15} + \cdots + 540) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 24 q^{5} - 24 q^{7} + 540 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 24 q^{5} - 24 q^{7} + 540 q^{9} + 372 q^{11} - 224 q^{13} + 480 q^{14} - 252 q^{15} - 2328 q^{16} - 840 q^{19} + 228 q^{20} + 936 q^{21} + 3536 q^{22} - 1404 q^{24} - 828 q^{26} - 1984 q^{28} - 5064 q^{29} + 1712 q^{31} - 7260 q^{32} - 540 q^{33} + 8040 q^{34} + 5976 q^{35} - 868 q^{37} - 2952 q^{39} + 6336 q^{40} - 6000 q^{41} + 648 q^{42} + 20868 q^{44} + 648 q^{45} - 1992 q^{46} + 1692 q^{47} + 2592 q^{48} + 8952 q^{50} - 12416 q^{52} - 18816 q^{53} - 3280 q^{55} - 9360 q^{57} + 1256 q^{58} - 6588 q^{59} + 9036 q^{60} - 5512 q^{61} - 648 q^{63} + 33048 q^{65} + 13464 q^{66} - 4704 q^{67} - 28296 q^{68} - 31096 q^{70} + 9756 q^{71} + 13692 q^{73} - 28296 q^{74} - 17072 q^{76} - 23868 q^{78} - 5056 q^{79} + 42228 q^{80} + 14580 q^{81} + 24708 q^{83} + 26496 q^{84} - 4752 q^{85} - 77424 q^{86} + 5256 q^{87} + 36000 q^{89} + 19104 q^{91} + 12912 q^{92} - 18432 q^{93} + 112832 q^{94} - 52344 q^{96} + 50012 q^{97} + 11544 q^{98} + 10044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 82\!\cdots\!67 \nu^{19} + \cdots + 11\!\cdots\!60 ) / 56\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\!\cdots\!04 \nu^{19} + \cdots - 15\!\cdots\!68 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!35 \nu^{19} + \cdots - 14\!\cdots\!52 ) / 11\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47\!\cdots\!25 \nu^{19} + \cdots - 13\!\cdots\!72 ) / 50\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26\!\cdots\!19 \nu^{19} + \cdots + 10\!\cdots\!12 ) / 21\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22\!\cdots\!41 \nu^{19} + \cdots - 50\!\cdots\!12 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!12 \nu^{19} + \cdots + 58\!\cdots\!32 ) / 50\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!27 \nu^{19} + \cdots - 21\!\cdots\!32 ) / 32\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 74\!\cdots\!63 \nu^{19} + \cdots - 14\!\cdots\!68 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!77 \nu^{19} + \cdots + 44\!\cdots\!76 ) / 12\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 88\!\cdots\!41 \nu^{19} + \cdots - 13\!\cdots\!44 ) / 48\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 33\!\cdots\!81 \nu^{19} + \cdots - 92\!\cdots\!00 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15\!\cdots\!84 \nu^{19} + \cdots - 54\!\cdots\!12 ) / 67\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 32\!\cdots\!87 \nu^{19} + \cdots + 44\!\cdots\!84 ) / 11\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 15\!\cdots\!56 \nu^{19} + \cdots - 48\!\cdots\!44 ) / 38\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 12\!\cdots\!27 \nu^{19} + \cdots - 35\!\cdots\!28 ) / 29\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 50\!\cdots\!67 \nu^{19} + \cdots + 14\!\cdots\!56 ) / 81\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 13\!\cdots\!99 \nu^{19} + \cdots + 18\!\cdots\!48 ) / 11\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{9} + 26\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - \beta_{14} - 3\beta_{9} + 3\beta_{6} + 42\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} + \beta_{14} - 3 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + \beta_{10} + 3 \beta_{8} + \cdots - 1113 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{19} + 71 \beta_{18} + 64 \beta_{17} + 9 \beta_{15} + 22 \beta_{13} + 6 \beta_{10} + 288 \beta_{9} + \cdots + 377 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 48 \beta_{19} - 18 \beta_{18} - 84 \beta_{17} + 18 \beta_{16} - 2843 \beta_{15} - 84 \beta_{14} + \cdots + 1456 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 612 \beta_{19} - 4327 \beta_{16} + 978 \beta_{15} + 3551 \beta_{14} + 2120 \beta_{12} - 2224 \beta_{11} + \cdots - 38660 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2178 \beta_{18} + 5693 \beta_{17} + 2178 \beta_{16} - 5693 \beta_{14} + 18093 \beta_{13} + \cdots + 2651299 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 45468 \beta_{19} - 253459 \beta_{18} - 192438 \beta_{17} - 77889 \beta_{15} - 168206 \beta_{13} + \cdots - 2930125 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4210 \beta_{19} + 192120 \beta_{18} + 363150 \beta_{17} - 192120 \beta_{16} + 7747399 \beta_{15} + \cdots - 7528124 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3000672 \beta_{19} + 14603735 \beta_{16} - 5580402 \beta_{15} - 10443265 \beta_{14} + \cdots + 198445014 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14864580 \beta_{18} - 22611365 \beta_{17} - 14864580 \beta_{16} + 22611365 \beta_{14} + \cdots - 7238942981 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 186968658 \beta_{19} + 834317419 \beta_{18} + 571043708 \beta_{17} + 380465133 \beta_{15} + \cdots + 12748032461 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 462121580 \beta_{19} - 1067886438 \beta_{18} - 1389237200 \beta_{17} + 1067886438 \beta_{16} + \cdots + 30090976712 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 11286969396 \beta_{19} - 47446397607 \beta_{16} + 25195294266 \beta_{15} + 31492895987 \beta_{14} + \cdots - 796558371952 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 73104401742 \beta_{18} + 84582095837 \beta_{17} + 73104401742 \beta_{16} - 84582095837 \beta_{14} + \cdots + 21249258265383 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 668742068952 \beta_{19} - 2691935807403 \beta_{18} - 1750626100362 \beta_{17} - 1635366242241 \beta_{15} + \cdots - 48995546053117 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2746950575330 \beta_{19} + 4838343346236 \beta_{18} + 5114695060562 \beta_{17} + \cdots - 112009570267140 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 39167603966496 \beta_{19} + 152593590865935 \beta_{16} - 104536765461618 \beta_{15} + \cdots + 29\!\cdots\!86 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
5.10926 − 5.10926i
4.32919 − 4.32919i
4.05053 − 4.05053i
1.99524 − 1.99524i
−0.0868646 + 0.0868646i
−0.755803 + 0.755803i
−1.66937 + 1.66937i
−2.89776 + 2.89776i
−4.67933 + 4.67933i
−5.39509 + 5.39509i
5.10926 + 5.10926i
4.32919 + 4.32919i
4.05053 + 4.05053i
1.99524 + 1.99524i
−0.0868646 − 0.0868646i
−0.755803 − 0.755803i
−1.66937 − 1.66937i
−2.89776 − 2.89776i
−4.67933 − 4.67933i
−5.39509 − 5.39509i
−5.10926 + 5.10926i −5.19615 − 36.2092i 3.64937 − 3.64937i 26.5485 − 26.5485i −17.1355 − 17.1355i 103.254 + 103.254i 27.0000 37.2912i
31.2 −4.32919 + 4.32919i 5.19615 − 21.4837i −26.0804 + 26.0804i −22.4951 + 22.4951i −16.9579 − 16.9579i 23.7399 + 23.7399i 27.0000 − 225.814i
31.3 −4.05053 + 4.05053i 5.19615 − 16.8136i 33.5573 − 33.5573i −21.0472 + 21.0472i 23.7132 + 23.7132i 3.29565 + 3.29565i 27.0000 271.850i
31.4 −1.99524 + 1.99524i −5.19615 8.03802i 3.82538 − 3.82538i 10.3676 − 10.3676i −42.4180 − 42.4180i −47.9617 − 47.9617i 27.0000 15.2651i
31.5 0.0868646 − 0.0868646i 5.19615 15.9849i −14.6130 + 14.6130i 0.451362 − 0.451362i 21.4788 + 21.4788i 2.77836 + 2.77836i 27.0000 2.53870i
31.6 0.755803 − 0.755803i −5.19615 14.8575i 25.9238 − 25.9238i −3.92727 + 3.92727i 55.8718 + 55.8718i 23.3222 + 23.3222i 27.0000 − 39.1866i
31.7 1.66937 − 1.66937i −5.19615 10.4264i −28.4160 + 28.4160i −8.67431 + 8.67431i −17.1341 − 17.1341i 44.1155 + 44.1155i 27.0000 94.8736i
31.8 2.89776 − 2.89776i 5.19615 − 0.794047i 18.6373 − 18.6373i 15.0572 − 15.0572i −15.2549 − 15.2549i 44.0632 + 44.0632i 27.0000 − 108.013i
31.9 4.67933 − 4.67933i −5.19615 − 27.7923i 13.1418 − 13.1418i −24.3145 + 24.3145i −30.2175 − 30.2175i −55.1800 − 55.1800i 27.0000 − 122.989i
31.10 5.39509 − 5.39509i 5.19615 − 42.2140i −17.6256 + 17.6256i 28.0337 − 28.0337i 26.0542 + 26.0542i −141.427 − 141.427i 27.0000 190.184i
34.1 −5.10926 − 5.10926i −5.19615 36.2092i 3.64937 + 3.64937i 26.5485 + 26.5485i −17.1355 + 17.1355i 103.254 − 103.254i 27.0000 − 37.2912i
34.2 −4.32919 − 4.32919i 5.19615 21.4837i −26.0804 − 26.0804i −22.4951 − 22.4951i −16.9579 + 16.9579i 23.7399 − 23.7399i 27.0000 225.814i
34.3 −4.05053 − 4.05053i 5.19615 16.8136i 33.5573 + 33.5573i −21.0472 − 21.0472i 23.7132 − 23.7132i 3.29565 − 3.29565i 27.0000 − 271.850i
34.4 −1.99524 − 1.99524i −5.19615 − 8.03802i 3.82538 + 3.82538i 10.3676 + 10.3676i −42.4180 + 42.4180i −47.9617 + 47.9617i 27.0000 − 15.2651i
34.5 0.0868646 + 0.0868646i 5.19615 − 15.9849i −14.6130 − 14.6130i 0.451362 + 0.451362i 21.4788 − 21.4788i 2.77836 − 2.77836i 27.0000 − 2.53870i
34.6 0.755803 + 0.755803i −5.19615 − 14.8575i 25.9238 + 25.9238i −3.92727 − 3.92727i 55.8718 − 55.8718i 23.3222 − 23.3222i 27.0000 39.1866i
34.7 1.66937 + 1.66937i −5.19615 − 10.4264i −28.4160 − 28.4160i −8.67431 − 8.67431i −17.1341 + 17.1341i 44.1155 − 44.1155i 27.0000 − 94.8736i
34.8 2.89776 + 2.89776i 5.19615 0.794047i 18.6373 + 18.6373i 15.0572 + 15.0572i −15.2549 + 15.2549i 44.0632 − 44.0632i 27.0000 108.013i
34.9 4.67933 + 4.67933i −5.19615 27.7923i 13.1418 + 13.1418i −24.3145 − 24.3145i −30.2175 + 30.2175i −55.1800 + 55.1800i 27.0000 122.989i
34.10 5.39509 + 5.39509i 5.19615 42.2140i −17.6256 − 17.6256i 28.0337 + 28.0337i 26.0542 − 26.0542i −141.427 + 141.427i 27.0000 − 190.184i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.5.g.a ✓ 20
3.b odd 2 1 117.5.j.b 20
13.d odd 4 1 inner 39.5.g.a ✓ 20
39.f even 4 1 117.5.j.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.5.g.a ✓ 20 1.a even 1 1 trivial
39.5.g.a ✓ 20 13.d odd 4 1 inner
117.5.j.b 20 3.b odd 2 1
117.5.j.b 20 39.f even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 2103506496 \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 53\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 84\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 42\!\cdots\!64)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 10\!\cdots\!52)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 71\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
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