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Properties

Label 39.3.h.b
Level $39$
Weight $3$
Character orbit 39.h
Analytic conductor $1.063$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [39,3,Mod(17,39)]
 
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([3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 39.h (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: \(1.06267303101\)
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: \( x^{12} - x^{10} - 12x^{9} - 2x^{8} + 6x^{7} + 87x^{6} + 18x^{5} - 18x^{4} - 324x^{3} - 81x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{11} q^{3} + ( - \beta_{6} + 3 \beta_{4} - \beta_{2} + \cdots - 3) q^{4}+ \cdots + ( - \beta_{11} - \beta_{10} - \beta_{9} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_{11} q^{3} + ( - \beta_{6} + 3 \beta_{4} - \beta_{2} + \cdots - 3) q^{4}+ \cdots + (24 \beta_{11} - 5 \beta_{9} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} - 14 q^{4} + 6 q^{6} - 36 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} - 14 q^{4} + 6 q^{6} - 36 q^{7} + 10 q^{9} - 14 q^{10} + 76 q^{12} + 2 q^{13} - 60 q^{15} + 34 q^{16} - 12 q^{19} - 20 q^{22} - 78 q^{24} - 80 q^{25} - 76 q^{27} + 192 q^{28} + 36 q^{30} + 210 q^{33} + 86 q^{36} + 270 q^{37} - 128 q^{39} - 172 q^{40} - 150 q^{42} + 48 q^{43} - 168 q^{45} - 300 q^{46} - 34 q^{48} - 126 q^{49} + 228 q^{51} + 80 q^{52} - 510 q^{54} + 76 q^{55} + 294 q^{58} + 142 q^{61} + 12 q^{63} - 244 q^{64} + 612 q^{66} + 264 q^{67} + 42 q^{69} + 480 q^{72} + 220 q^{75} - 216 q^{76} - 128 q^{79} + 58 q^{81} - 182 q^{82} - 462 q^{84} - 618 q^{85} - 348 q^{87} - 592 q^{88} + 348 q^{90} - 204 q^{91} - 576 q^{93} + 448 q^{94} + 372 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{10} - 12x^{9} - 2x^{8} + 6x^{7} + 87x^{6} + 18x^{5} - 18x^{4} - 324x^{3} - 81x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + \nu^{8} + 12\nu^{7} + 2\nu^{6} - 6\nu^{5} - 87\nu^{4} - 18\nu^{3} + 18\nu^{2} + 324\nu + 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{11} + 18 \nu^{10} + 56 \nu^{9} + 168 \nu^{8} + 247 \nu^{7} - 534 \nu^{6} - 615 \nu^{5} + \cdots - 4374 ) / 3159 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6 \nu^{11} + 2 \nu^{10} + 12 \nu^{9} - 29 \nu^{8} - 94 \nu^{6} + 66 \nu^{5} - 78 \nu^{4} + 360 \nu^{3} + \cdots + 567 ) / 1053 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23 \nu^{11} - 12 \nu^{10} + 32 \nu^{9} + 369 \nu^{8} + 208 \nu^{7} - 60 \nu^{6} - 1956 \nu^{5} + \cdots - 243 ) / 3159 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8 \nu^{11} + 6 \nu^{10} + 62 \nu^{9} + 108 \nu^{8} + 52 \nu^{7} - 321 \nu^{6} - 660 \nu^{5} + \cdots + 648 ) / 1053 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29 \nu^{11} - 66 \nu^{10} - 97 \nu^{9} + 333 \nu^{8} + 598 \nu^{7} + 1308 \nu^{6} - 1398 \nu^{5} + \cdots + 22356 ) / 3159 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} - 4 \nu^{10} - 2 \nu^{9} + 7 \nu^{8} + 35 \nu^{7} + 47 \nu^{6} - 6 \nu^{5} - 150 \nu^{4} + \cdots + 567 ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} + 5 \nu^{9} - 14 \nu^{8} - 23 \nu^{7} - 70 \nu^{6} + 51 \nu^{5} + 120 \nu^{4} + \cdots - 1053 ) / 81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 43 \nu^{11} + 36 \nu^{10} + 164 \nu^{9} - 444 \nu^{8} - 806 \nu^{7} - 2082 \nu^{6} + 2085 \nu^{5} + \cdots - 27702 ) / 3159 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2 \nu^{11} + 5 \nu^{10} + 4 \nu^{9} - 14 \nu^{8} - 52 \nu^{7} - 40 \nu^{6} + 48 \nu^{5} + 234 \nu^{4} + \cdots - 162 ) / 117 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 2 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} + \cdots + 18 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11 \beta_{11} + 11 \beta_{10} - 7 \beta_{9} + 7 \beta_{8} - \beta_{7} - 7 \beta_{6} - 2 \beta_{5} + \cdots + 18 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 4 \beta_{7} - \beta_{6} + 4 \beta_{5} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 26 \beta_{11} - 23 \beta_{10} + 10 \beta_{9} + 5 \beta_{8} - 14 \beta_{7} - 5 \beta_{6} + \cdots + 10 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12 \beta_{11} + 12 \beta_{10} - 6 \beta_{9} + 6 \beta_{8} + 6 \beta_{7} - 12 \beta_{6} + 12 \beta_{5} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 143 \beta_{11} - 70 \beta_{10} + 71 \beta_{9} + 142 \beta_{8} - 13 \beta_{7} + 74 \beta_{6} + 13 \beta_{5} + \cdots - 504 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 25 \beta_{11} - 14 \beta_{10} - 44 \beta_{9} - 22 \beta_{8} - 80 \beta_{7} + 22 \beta_{6} + \cdots - 44 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 169 \beta_{11} + 169 \beta_{10} - 83 \beta_{9} + 83 \beta_{8} + 235 \beta_{7} - 407 \beta_{6} + \cdots - 1314 ) / 6 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−1.41341 + 1.00114i
−0.528217 − 1.64954i
1.66216 + 0.487054i
−1.25288 − 1.19595i
1.69265 − 0.367321i
−0.160307 + 1.72462i
−1.41341 − 1.00114i
−0.528217 + 1.64954i
1.66216 − 0.487054i
−1.25288 + 1.19595i
1.69265 + 0.367321i
−0.160307 − 1.72462i
−1.65124 + 2.86002i −2.94860 + 0.552936i −3.45316 − 5.98105i 0.796269 3.28743 − 9.34610i −7.72115 + 4.45781i 9.59803 8.38852 − 3.26078i −1.31483 + 2.27735i
17.2 −1.26725 + 2.19493i 2.73015 + 1.24349i −1.21182 − 2.09894i −1.90725 −6.18916 + 4.41669i 0.493308 − 0.284812i −3.99526 5.90744 + 6.78985i 2.41695 − 4.18629i
17.3 −0.646150 + 1.11916i 0.139425 − 2.99676i 1.16498 + 2.01781i 7.12238 3.26377 + 2.09239i −1.77216 + 1.02316i −8.18021 −8.96112 − 0.835646i −4.60213 + 7.97112i
17.4 0.646150 − 1.11916i 2.52556 − 1.61912i 1.16498 + 2.01781i −7.12238 −0.180179 − 3.87271i −1.77216 + 1.02316i 8.18021 3.75687 − 8.17838i −4.60213 + 7.97112i
17.5 1.26725 − 2.19493i −2.44197 − 1.74263i −1.21182 − 2.09894i 1.90725 −6.91954 + 3.15163i 0.493308 − 0.284812i 3.99526 2.92647 + 8.51092i 2.41695 − 4.18629i
17.6 1.65124 − 2.86002i 0.995445 + 2.83003i −3.45316 − 5.98105i −0.796269 9.73768 + 1.82606i −7.72115 + 4.45781i −9.59803 −7.01818 + 5.63428i −1.31483 + 2.27735i
23.1 −1.65124 − 2.86002i −2.94860 − 0.552936i −3.45316 + 5.98105i 0.796269 3.28743 + 9.34610i −7.72115 − 4.45781i 9.59803 8.38852 + 3.26078i −1.31483 − 2.27735i
23.2 −1.26725 − 2.19493i 2.73015 − 1.24349i −1.21182 + 2.09894i −1.90725 −6.18916 − 4.41669i 0.493308 + 0.284812i −3.99526 5.90744 − 6.78985i 2.41695 + 4.18629i
23.3 −0.646150 − 1.11916i 0.139425 + 2.99676i 1.16498 − 2.01781i 7.12238 3.26377 − 2.09239i −1.77216 − 1.02316i −8.18021 −8.96112 + 0.835646i −4.60213 − 7.97112i
23.4 0.646150 + 1.11916i 2.52556 + 1.61912i 1.16498 − 2.01781i −7.12238 −0.180179 + 3.87271i −1.77216 − 1.02316i 8.18021 3.75687 + 8.17838i −4.60213 − 7.97112i
23.5 1.26725 + 2.19493i −2.44197 + 1.74263i −1.21182 + 2.09894i 1.90725 −6.91954 − 3.15163i 0.493308 + 0.284812i 3.99526 2.92647 − 8.51092i 2.41695 + 4.18629i
23.6 1.65124 + 2.86002i 0.995445 − 2.83003i −3.45316 + 5.98105i −0.796269 9.73768 − 1.82606i −7.72115 − 4.45781i −9.59803 −7.01818 − 5.63428i −1.31483 − 2.27735i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.3.h.b ✓ 12
3.b odd 2 1 inner 39.3.h.b ✓ 12
13.c even 3 1 507.3.d.c 12
13.e even 6 1 inner 39.3.h.b ✓ 12
13.e even 6 1 507.3.d.c 12
13.f odd 12 2 507.3.c.j 12
39.h odd 6 1 inner 39.3.h.b ✓ 12
39.h odd 6 1 507.3.d.c 12
39.i odd 6 1 507.3.d.c 12
39.k even 12 2 507.3.c.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.3.h.b ✓ 12 1.a even 1 1 trivial
39.3.h.b ✓ 12 3.b odd 2 1 inner
39.3.h.b ✓ 12 13.e even 6 1 inner
39.3.h.b ✓ 12 39.h odd 6 1 inner
507.3.c.j 12 13.f odd 12 2
507.3.c.j 12 39.k even 12 2
507.3.d.c 12 13.c even 3 1
507.3.d.c 12 13.e even 6 1
507.3.d.c 12 39.h odd 6 1
507.3.d.c 12 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 19T_{2}^{10} + 262T_{2}^{8} + 1647T_{2}^{6} + 7578T_{2}^{4} + 11583T_{2}^{2} + 13689 \) acting on \(S_{3}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 19 T^{10} + \cdots + 13689 \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{6} - 55 T^{4} + \cdots - 117)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 18 T^{5} + \cdots + 108)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 22791725268624 \) Copy content Toggle raw display
$13$ \( (T^{6} - T^{5} + \cdots + 4826809)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 285018244641 \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + \cdots + 10178892)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 4560291914256 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 49\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( (T^{6} + 3756 T^{4} + \cdots + 289768752)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 135 T^{5} + \cdots + 19645443)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 4031409418569 \) Copy content Toggle raw display
$43$ \( (T^{6} - 24 T^{5} + \cdots + 70756)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 3556 T^{4} + \cdots - 122204628)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 10341 T^{4} + \cdots + 29987000199)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{6} - 71 T^{5} + \cdots + 111188235601)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 132 T^{5} + \cdots + 11371117068)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{6} + 9177 T^{4} + \cdots + 887416803)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 32 T^{2} + \cdots + 5968)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 26056 T^{4} + \cdots - 119314301652)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{6} - 186 T^{5} + \cdots + 434595888)^{2} \) Copy content Toggle raw display
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