Properties

Label 39.3.d.c
Level $39$
Weight $3$
Character orbit 39.d
Analytic conductor $1.063$
Analytic rank $0$
Dimension $4$
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(38,39)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.38");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 39.d (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: \(1.06267303101\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-35})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 13x^{2} - 12x + 141 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{3} - q^{4} - 3 \beta_{2} q^{5} - \beta_{3} q^{6} + (2 \beta_{3} + \beta_{2}) q^{7} + 5 \beta_{2} q^{8} + ( - \beta_1 - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{3} - q^{4} - 3 \beta_{2} q^{5} - \beta_{3} q^{6} + (2 \beta_{3} + \beta_{2}) q^{7} + 5 \beta_{2} q^{8} + ( - \beta_1 - 8) q^{9} + 9 q^{10} + 2 \beta_{2} q^{11} + ( - \beta_1 + 1) q^{12} + ( - 2 \beta_{3} - \beta_{2} + 8) q^{13} + ( - 6 \beta_1 + 3) q^{14} - 3 \beta_{3} q^{15} - 11 q^{16} + (6 \beta_1 - 3) q^{17} + (\beta_{3} + 9 \beta_{2}) q^{18} + (4 \beta_{3} + 2 \beta_{2}) q^{19} + 3 \beta_{2} q^{20} + ( - \beta_{3} - 18 \beta_{2}) q^{21} - 6 q^{22} + 5 \beta_{3} q^{24} + 2 q^{25} + ( - 8 \beta_{2} + 6 \beta_1 - 3) q^{26} + ( - 8 \beta_1 + 17) q^{27} + ( - 2 \beta_{3} - \beta_{2}) q^{28} + (12 \beta_1 - 6) q^{29} + (9 \beta_1 - 9) q^{30} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{31} - 9 \beta_{2} q^{32} + 2 \beta_{3} q^{33} + ( - 6 \beta_{3} - 3 \beta_{2}) q^{34} + ( - 18 \beta_1 + 9) q^{35} + (\beta_1 + 8) q^{36} + (2 \beta_{3} + \beta_{2}) q^{37} + ( - 12 \beta_1 + 6) q^{38} + (\beta_{3} + 18 \beta_{2} + 8 \beta_1 - 8) q^{39} - 45 q^{40} - 2 \beta_{2} q^{41} + (3 \beta_1 + 51) q^{42} + 35 q^{43} - 2 \beta_{2} q^{44} + (3 \beta_{3} + 27 \beta_{2}) q^{45} + 31 \beta_{2} q^{47} + ( - 11 \beta_1 + 11) q^{48} - 56 q^{49} - 2 \beta_{2} q^{50} + ( - 3 \beta_1 - 51) q^{51} + (2 \beta_{3} + \beta_{2} - 8) q^{52} + ( - 24 \beta_1 + 12) q^{53} + (8 \beta_{3} - 9 \beta_{2}) q^{54} - 18 q^{55} + (30 \beta_1 - 15) q^{56} + ( - 2 \beta_{3} - 36 \beta_{2}) q^{57} + ( - 12 \beta_{3} - 6 \beta_{2}) q^{58} + 12 \beta_{2} q^{59} + 3 \beta_{3} q^{60} + 70 q^{61} + (12 \beta_1 - 6) q^{62} + ( - 17 \beta_{3} + 9 \beta_{2}) q^{63} + 71 q^{64} + ( - 24 \beta_{2} + 18 \beta_1 - 9) q^{65} + ( - 6 \beta_1 + 6) q^{66} + ( - 6 \beta_1 + 3) q^{68} + (18 \beta_{3} + 9 \beta_{2}) q^{70} - 47 \beta_{2} q^{71} + ( - 5 \beta_{3} - 45 \beta_{2}) q^{72} + (24 \beta_{3} + 12 \beta_{2}) q^{73} + ( - 6 \beta_1 + 3) q^{74} + (2 \beta_1 - 2) q^{75} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{76} + (12 \beta_1 - 6) q^{77} + ( - 8 \beta_{3} - 3 \beta_1 - 51) q^{78} - 50 q^{79} + 33 \beta_{2} q^{80} + (17 \beta_1 + 55) q^{81} + 6 q^{82} + 74 \beta_{2} q^{83} + (\beta_{3} + 18 \beta_{2}) q^{84} + ( - 18 \beta_{3} - 9 \beta_{2}) q^{85} - 35 \beta_{2} q^{86} + ( - 6 \beta_1 - 102) q^{87} + 30 q^{88} - 62 \beta_{2} q^{89} + ( - 9 \beta_1 - 72) q^{90} + (16 \beta_{3} + 8 \beta_{2} + 105) q^{91} + (2 \beta_{3} + 36 \beta_{2}) q^{93} - 93 q^{94} + ( - 36 \beta_1 + 18) q^{95} - 9 \beta_{3} q^{96} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{97} + 56 \beta_{2} q^{98} + ( - 2 \beta_{3} - 18 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{4} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{4} - 34 q^{9} + 36 q^{10} + 2 q^{12} + 32 q^{13} - 44 q^{16} - 24 q^{22} + 8 q^{25} + 52 q^{27} - 18 q^{30} + 34 q^{36} - 16 q^{39} - 180 q^{40} + 210 q^{42} + 140 q^{43} + 22 q^{48} - 224 q^{49} - 210 q^{51} - 32 q^{52} - 72 q^{55} + 280 q^{61} + 284 q^{64} + 12 q^{66} - 4 q^{75} - 210 q^{78} - 200 q^{79} + 254 q^{81} + 24 q^{82} - 420 q^{87} + 120 q^{88} - 306 q^{90} + 420 q^{91} - 372 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 13x^{2} - 12x + 141 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} + 48\nu ) / 47 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} - \nu ) / 47 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 22\nu^{2} - 23\nu + 141 ) / 47 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} - 21\beta_{2} + \beta _1 - 9 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1
2.23205 2.95804i
2.23205 + 2.95804i
−1.23205 2.95804i
−1.23205 + 2.95804i
−1.73205 −0.500000 2.95804i −1.00000 −5.19615 0.866025 + 5.12348i 10.2470i 8.66025 −8.50000 + 2.95804i 9.00000
38.2 −1.73205 −0.500000 + 2.95804i −1.00000 −5.19615 0.866025 5.12348i 10.2470i 8.66025 −8.50000 2.95804i 9.00000
38.3 1.73205 −0.500000 2.95804i −1.00000 5.19615 −0.866025 5.12348i 10.2470i −8.66025 −8.50000 + 2.95804i 9.00000
38.4 1.73205 −0.500000 + 2.95804i −1.00000 5.19615 −0.866025 + 5.12348i 10.2470i −8.66025 −8.50000 2.95804i 9.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.3.d.c 4
3.b odd 2 1 inner 39.3.d.c 4
4.b odd 2 1 624.3.l.f 4
12.b even 2 1 624.3.l.f 4
13.b even 2 1 inner 39.3.d.c 4
13.d odd 4 2 507.3.c.f 4
39.d odd 2 1 inner 39.3.d.c 4
39.f even 4 2 507.3.c.f 4
52.b odd 2 1 624.3.l.f 4
156.h even 2 1 624.3.l.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.3.d.c 4 1.a even 1 1 trivial
39.3.d.c 4 3.b odd 2 1 inner
39.3.d.c 4 13.b even 2 1 inner
39.3.d.c 4 39.d odd 2 1 inner
507.3.c.f 4 13.d odd 4 2
507.3.c.f 4 39.f even 4 2
624.3.l.f 4 4.b odd 2 1
624.3.l.f 4 12.b even 2 1
624.3.l.f 4 52.b odd 2 1
624.3.l.f 4 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(39, [\chi])\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{2} - 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 105)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 16 T + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 315)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 420)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1260)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 420)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 105)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T - 35)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 2883)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 5040)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$61$ \( (T - 70)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 6627)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 15120)^{2} \) Copy content Toggle raw display
$79$ \( (T + 50)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 16428)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 11532)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 420)^{2} \) Copy content Toggle raw display
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