Properties

Label 3822.2.c.h
Level $3822$
Weight $2$
Character orbit 3822.c
Analytic conductor $30.519$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3822,2,Mod(883,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3822.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + q^{3} - q^{4} + ( - \beta_{2} + \beta_1) q^{5} + \beta_{2} q^{6} - \beta_{2} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + q^{3} - q^{4} + ( - \beta_{2} + \beta_1) q^{5} + \beta_{2} q^{6} - \beta_{2} q^{8} + q^{9} + ( - \beta_{3} + 2) q^{10} + (3 \beta_{2} - \beta_1) q^{11} - q^{12} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{13} + ( - \beta_{2} + \beta_1) q^{15} + q^{16} + ( - 3 \beta_{3} + 2) q^{17} + \beta_{2} q^{18} + ( - \beta_{2} - \beta_1) q^{19} + (\beta_{2} - \beta_1) q^{20} + (\beta_{3} - 4) q^{22} + (3 \beta_{3} - 2) q^{23} - \beta_{2} q^{24} + (3 \beta_{3} - 3) q^{25} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{26} + q^{27} - \beta_{3} q^{29} + ( - \beta_{3} + 2) q^{30} + ( - 4 \beta_{2} - 4 \beta_1) q^{31} + \beta_{2} q^{32} + (3 \beta_{2} - \beta_1) q^{33} + ( - \beta_{2} - 3 \beta_1) q^{34} - q^{36} + (3 \beta_{2} - 3 \beta_1) q^{37} + \beta_{3} q^{38} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{39} + (\beta_{3} - 2) q^{40} - 4 \beta_{2} q^{41} + ( - \beta_{3} - 8) q^{43} + ( - 3 \beta_{2} + \beta_1) q^{44} + ( - \beta_{2} + \beta_1) q^{45} + (\beta_{2} + 3 \beta_1) q^{46} - 4 \beta_1 q^{47} + q^{48} + 3 \beta_1 q^{50} + ( - 3 \beta_{3} + 2) q^{51} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{52} + (2 \beta_{3} + 8) q^{53} + \beta_{2} q^{54} + ( - 5 \beta_{3} + 12) q^{55} + ( - \beta_{2} - \beta_1) q^{57} + ( - \beta_{2} - \beta_1) q^{58} + (6 \beta_{2} + 4 \beta_1) q^{59} + (\beta_{2} - \beta_1) q^{60} - \beta_{3} q^{61} + 4 \beta_{3} q^{62} - q^{64} + (5 \beta_{2} - 3 \beta_1 + 2) q^{65} + (\beta_{3} - 4) q^{66} + (4 \beta_{2} + 2 \beta_1) q^{67} + (3 \beta_{3} - 2) q^{68} + (3 \beta_{3} - 2) q^{69} + ( - 6 \beta_{2} - 6 \beta_1) q^{71} - \beta_{2} q^{72} + ( - 9 \beta_{2} + \beta_1) q^{73} + (3 \beta_{3} - 6) q^{74} + (3 \beta_{3} - 3) q^{75} + (\beta_{2} + \beta_1) q^{76} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{78} + 16 q^{79} + ( - \beta_{2} + \beta_1) q^{80} + q^{81} + 4 q^{82} + 2 \beta_{2} q^{83} + ( - 11 \beta_{2} + 5 \beta_1) q^{85} + ( - 9 \beta_{2} - \beta_1) q^{86} - \beta_{3} q^{87} + ( - \beta_{3} + 4) q^{88} + 8 \beta_{2} q^{89} + ( - \beta_{3} + 2) q^{90} + ( - 3 \beta_{3} + 2) q^{92} + ( - 4 \beta_{2} - 4 \beta_1) q^{93} + (4 \beta_{3} - 4) q^{94} + ( - \beta_{3} + 4) q^{95} + \beta_{2} q^{96} - 10 \beta_{2} q^{97} + (3 \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 6 q^{10} - 4 q^{12} - 6 q^{13} + 4 q^{16} + 2 q^{17} - 14 q^{22} - 2 q^{23} - 6 q^{25} + 6 q^{26} + 4 q^{27} - 2 q^{29} + 6 q^{30} - 4 q^{36} + 2 q^{38} - 6 q^{39} - 6 q^{40} - 34 q^{43} + 4 q^{48} + 2 q^{51} + 6 q^{52} + 36 q^{53} + 38 q^{55} - 2 q^{61} + 8 q^{62} - 4 q^{64} + 8 q^{65} - 14 q^{66} - 2 q^{68} - 2 q^{69} - 18 q^{74} - 6 q^{75} + 6 q^{78} + 64 q^{79} + 4 q^{81} + 16 q^{82} - 2 q^{87} + 14 q^{88} + 6 q^{90} + 2 q^{92} - 8 q^{94} + 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times\).

\(n\) \(1471\) \(2549\) \(3433\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
1.56155i
2.56155i
2.56155i
1.56155i
1.00000i 1.00000 −1.00000 0.561553i 1.00000i 0 1.00000i 1.00000 −0.561553
883.2 1.00000i 1.00000 −1.00000 3.56155i 1.00000i 0 1.00000i 1.00000 3.56155
883.3 1.00000i 1.00000 −1.00000 3.56155i 1.00000i 0 1.00000i 1.00000 3.56155
883.4 1.00000i 1.00000 −1.00000 0.561553i 1.00000i 0 1.00000i 1.00000 −0.561553
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3822.2.c.h 4
7.b odd 2 1 546.2.c.e 4
13.b even 2 1 inner 3822.2.c.h 4
21.c even 2 1 1638.2.c.h 4
28.d even 2 1 4368.2.h.n 4
91.b odd 2 1 546.2.c.e 4
91.i even 4 1 7098.2.a.bg 2
91.i even 4 1 7098.2.a.bv 2
273.g even 2 1 1638.2.c.h 4
364.h even 2 1 4368.2.h.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.e 4 7.b odd 2 1
546.2.c.e 4 91.b odd 2 1
1638.2.c.h 4 21.c even 2 1
1638.2.c.h 4 273.g even 2 1
3822.2.c.h 4 1.a even 1 1 trivial
3822.2.c.h 4 13.b even 2 1 inner
4368.2.h.n 4 28.d even 2 1
4368.2.h.n 4 364.h even 2 1
7098.2.a.bg 2 91.i even 4 1
7098.2.a.bv 2 91.i even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3822, [\chi])\):

\( T_{5}^{4} + 13T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 33T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{2} - T_{17} - 38 \) Copy content Toggle raw display
\( T_{19}^{4} + 9T_{19}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 33T^{2} + 64 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + 18 T^{2} + 78 T + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - T - 38)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + T - 38)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
$37$ \( T^{4} + 117T^{2} + 324 \) Copy content Toggle raw display
$41$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 17 T + 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( (T^{2} - 18 T + 64)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 168T^{2} + 2704 \) Copy content Toggle raw display
$61$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 52T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{4} + 324 T^{2} + 20736 \) Copy content Toggle raw display
$73$ \( T^{4} + 189T^{2} + 7396 \) Copy content Toggle raw display
$79$ \( (T - 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
show more
show less