Properties

Label 387.2.h.d
Level $387$
Weight $2$
Character orbit 387.h
Analytic conductor $3.090$
Analytic rank $0$
Dimension $4$
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] = [387,2,Mod(208,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.h (of order \(3\), 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: \(3.09021055822\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} - 3 \beta_{2} q^{4} - 2 \beta_1 q^{5} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{7} + ( - 4 \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} - 3 \beta_{2} q^{4} - 2 \beta_1 q^{5} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{7} + ( - 4 \beta_{2} + 1) q^{8} + ( - 2 \beta_{3} - 4 \beta_1 - 2) q^{10} + (\beta_{2} + 3) q^{11} + (2 \beta_{3} + \beta_{2} + \beta_1) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{14} + ( - 3 \beta_{2} + 5) q^{16} + ( - 3 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{17} - 2 \beta_1 q^{19} + ( - 6 \beta_{3} - 6 \beta_1 - 6) q^{20} + ( - \beta_{2} + 2) q^{22} + (5 \beta_{3} + \beta_1 + 5) q^{23} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{25} + (3 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{26} + (6 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{28} - 3 \beta_{3} q^{29} + ( - 3 \beta_{2} + 6) q^{32} + (2 \beta_{3} + 7 \beta_{2} + 7 \beta_1) q^{34} + (2 \beta_{2} + 4) q^{35} + (3 \beta_{3} - 3 \beta_1 + 3) q^{37} + ( - 2 \beta_{3} - 4 \beta_1 - 2) q^{38} + ( - 8 \beta_{3} - 10 \beta_1 - 8) q^{40} + ( - 4 \beta_{2} - 7) q^{41} + ( - \beta_{3} - 7) q^{43} + ( - 6 \beta_{2} - 3) q^{44} + (6 \beta_{3} + 7 \beta_1 + 6) q^{46} + (3 \beta_{2} - 3) q^{47} + ( - 6 \beta_{3} + 8 \beta_1 - 6) q^{49} + (3 \beta_{3} + 7 \beta_{2} + 7 \beta_1) q^{50} + (3 \beta_{3} + 9 \beta_{2} + 9 \beta_1) q^{52} + ( - 2 \beta_{3} - \beta_1 - 2) q^{53} + (2 \beta_{3} - 4 \beta_1 + 2) q^{55} + (5 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{56} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{58} + ( - 5 \beta_{2} - 2) q^{59} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{61} + ( - 6 \beta_{2} - 1) q^{64} + ( - 6 \beta_{2} + 2) q^{65} + (2 \beta_{3} - 3 \beta_1 + 2) q^{67} + (15 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{68} + 2 q^{70} + (4 \beta_{3} - 11 \beta_{2} - 11 \beta_1) q^{71} + ( - 3 \beta_{2} - 3 \beta_1) q^{73} - 3 \beta_1 q^{74} + ( - 6 \beta_{3} - 6 \beta_1 - 6) q^{76} + ( - 11 \beta_{3} + 7 \beta_{2} + 7 \beta_1) q^{77} + ( - 2 \beta_{3} - \beta_{2} - \beta_1) q^{79} + ( - 6 \beta_{3} - 16 \beta_1 - 6) q^{80} + ( - \beta_{2} - 3) q^{82} + (8 \beta_{3} - 13 \beta_1 + 8) q^{83} + ( - 4 \beta_{2} + 10) q^{85} + ( - \beta_{3} + 6 \beta_{2} - \beta_1 - 7) q^{86} + ( - 7 \beta_{2} - 1) q^{88} + 3 \beta_1 q^{89} + (4 \beta_{3} - 3 \beta_1 + 4) q^{91} + (3 \beta_{3} + 18 \beta_1 + 3) q^{92} + (9 \beta_{2} - 6) q^{94} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{95} + ( - 2 \beta_{2} - 8) q^{97} + (2 \beta_{3} + 10 \beta_1 + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 6 q^{4} - 2 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{2} + 6 q^{4} - 2 q^{5} + 4 q^{7} + 12 q^{8} - 8 q^{10} + 10 q^{11} - 5 q^{13} + q^{14} + 26 q^{16} + q^{17} - 2 q^{19} - 18 q^{20} + 10 q^{22} + 11 q^{23} - 2 q^{25} - 10 q^{26} - 9 q^{28} + 6 q^{29} + 30 q^{32} - 11 q^{34} + 12 q^{35} + 3 q^{37} - 8 q^{38} - 26 q^{40} - 20 q^{41} - 26 q^{43} + 19 q^{46} - 18 q^{47} - 4 q^{49} - 13 q^{50} - 15 q^{52} - 5 q^{53} - 8 q^{56} + 9 q^{58} + 2 q^{59} + q^{61} + 8 q^{64} + 20 q^{65} + q^{67} - 36 q^{68} + 8 q^{70} + 3 q^{71} + 3 q^{73} - 3 q^{74} - 18 q^{76} + 15 q^{77} + 5 q^{79} - 28 q^{80} - 10 q^{82} + 3 q^{83} + 48 q^{85} - 39 q^{86} + 10 q^{88} + 3 q^{89} + 5 q^{91} + 24 q^{92} - 42 q^{94} - 12 q^{95} - 28 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(173\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
−0.309017 0.535233i
0.809017 + 1.40126i
−0.309017 + 0.535233i
0.809017 1.40126i
0.381966 0 −1.85410 0.618034 + 1.07047i 0 2.11803 3.66854i −1.47214 0 0.236068 + 0.408882i
208.2 2.61803 0 4.85410 −1.61803 2.80252i 0 −0.118034 + 0.204441i 7.47214 0 −4.23607 7.33708i
307.1 0.381966 0 −1.85410 0.618034 1.07047i 0 2.11803 + 3.66854i −1.47214 0 0.236068 0.408882i
307.2 2.61803 0 4.85410 −1.61803 + 2.80252i 0 −0.118034 0.204441i 7.47214 0 −4.23607 + 7.33708i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.h.d 4
3.b odd 2 1 43.2.c.b 4
12.b even 2 1 688.2.i.e 4
43.c even 3 1 inner 387.2.h.d 4
129.f odd 6 1 43.2.c.b 4
129.f odd 6 1 1849.2.a.e 2
129.h even 6 1 1849.2.a.h 2
516.p even 6 1 688.2.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.b 4 3.b odd 2 1
43.2.c.b 4 129.f odd 6 1
387.2.h.d 4 1.a even 1 1 trivial
387.2.h.d 4 43.c even 3 1 inner
688.2.i.e 4 12.b even 2 1
688.2.i.e 4 516.p even 6 1
1849.2.a.e 2 129.f odd 6 1
1849.2.a.h 2 129.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + 17 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - 5 T + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} + 20 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} + 32 T^{2} + 31 T + 961 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 11 T^{3} + 92 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$41$ \( (T^{2} + 10 T + 5)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 13 T + 43)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9 T + 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 5 T^{3} + 20 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$59$ \( (T^{2} - T - 31)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - T^{3} + 12 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$67$ \( T^{4} - T^{3} + 12 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$71$ \( T^{4} - 3 T^{3} + 158 T^{2} + \cdots + 22201 \) Copy content Toggle raw display
$73$ \( T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$79$ \( T^{4} - 5 T^{3} + 20 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$83$ \( T^{4} - 3 T^{3} + 218 T^{2} + \cdots + 43681 \) Copy content Toggle raw display
$89$ \( T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T + 44)^{2} \) Copy content Toggle raw display
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