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Properties

Label 1872.2.c.j
Level $1872$
Weight $2$
Character orbit 1872.c
Analytic conductor $14.948$
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] = [1872,2,Mod(1585,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1872.1585");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.c (of order \(2\), degree \(1\), not 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: \(14.9479952584\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 104)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{2} + 2 \beta_1) q^{11} + (\beta_{3} - \beta_{2} - \beta_1 - 2) q^{13} + (\beta_{3} + 1) q^{17} - \beta_{2} q^{19} + ( - 2 \beta_{3} + 2) q^{23} + \beta_{3} q^{25} + 2 \beta_{3} q^{29} + (\beta_{2} + 2 \beta_1) q^{31} + (3 \beta_{3} - 7) q^{35} + (4 \beta_{2} + \beta_1) q^{37} + 4 \beta_{2} q^{41} + ( - \beta_{3} - 7) q^{43} + ( - 3 \beta_{2} - \beta_1) q^{47} + (5 \beta_{3} - 6) q^{49} + ( - 4 \beta_{3} - 2) q^{53} - 8 q^{55} + 5 \beta_{2} q^{59} + ( - 2 \beta_{3} + 8) q^{61} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{65} + (3 \beta_{2} + 2 \beta_1) q^{67} + (\beta_{2} + 3 \beta_1) q^{71} - 6 \beta_1 q^{73} + (4 \beta_{3} - 8) q^{77} + (2 \beta_{3} + 6) q^{79} + (\beta_{2} + 4 \beta_1) q^{83} + (2 \beta_{2} + \beta_1) q^{85} + 2 \beta_1 q^{89} + ( - \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 1) q^{91} + (2 \beta_{3} - 2) q^{95} + 4 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{13} + 6 q^{17} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 22 q^{35} - 30 q^{43} - 14 q^{49} - 16 q^{53} - 32 q^{55} + 28 q^{61} + 14 q^{65} - 24 q^{77} + 28 q^{79} + 2 q^{91} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
− 2.56155i
− 1.56155i
1.56155i
2.56155i
0 0 0 − 2.56155i 0 − 4.56155i 0 0 0
1585.2 0 0 0 − 1.56155i 0 0.438447i 0 0 0
1585.3 0 0 0 1.56155i 0 − 0.438447i 0 0 0
1585.4 0 0 0 2.56155i 0 4.56155i 0 0 0
\(n\): 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 1872.2.c.j 4
3.b odd 2 1 208.2.f.b 4
4.b odd 2 1 936.2.c.d 4
12.b even 2 1 104.2.f.a ✓ 4
13.b even 2 1 inner 1872.2.c.j 4
24.f even 2 1 832.2.f.i 4
24.h odd 2 1 832.2.f.f 4
39.d odd 2 1 208.2.f.b 4
39.f even 4 1 2704.2.a.s 2
39.f even 4 1 2704.2.a.t 2
52.b odd 2 1 936.2.c.d 4
60.h even 2 1 2600.2.k.a 4
60.l odd 4 1 2600.2.f.a 4
60.l odd 4 1 2600.2.f.b 4
156.h even 2 1 104.2.f.a ✓ 4
156.l odd 4 1 1352.2.a.d 2
156.l odd 4 1 1352.2.a.e 2
156.p even 6 2 1352.2.o.e 8
156.r even 6 2 1352.2.o.e 8
156.v odd 12 2 1352.2.i.g 4
156.v odd 12 2 1352.2.i.h 4
312.b odd 2 1 832.2.f.f 4
312.h even 2 1 832.2.f.i 4
780.d even 2 1 2600.2.k.a 4
780.w odd 4 1 2600.2.f.a 4
780.w odd 4 1 2600.2.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.f.a ✓ 4 12.b even 2 1
104.2.f.a ✓ 4 156.h even 2 1
208.2.f.b 4 3.b odd 2 1
208.2.f.b 4 39.d odd 2 1
832.2.f.f 4 24.h odd 2 1
832.2.f.f 4 312.b odd 2 1
832.2.f.i 4 24.f even 2 1
832.2.f.i 4 312.h even 2 1
936.2.c.d 4 4.b odd 2 1
936.2.c.d 4 52.b odd 2 1
1352.2.a.d 2 156.l odd 4 1
1352.2.a.e 2 156.l odd 4 1
1352.2.i.g 4 156.v odd 12 2
1352.2.i.h 4 156.v odd 12 2
1352.2.o.e 8 156.p even 6 2
1352.2.o.e 8 156.r even 6 2
1872.2.c.j 4 1.a even 1 1 trivial
1872.2.c.j 4 13.b even 2 1 inner
2600.2.f.a 4 60.l odd 4 1
2600.2.f.a 4 780.w odd 4 1
2600.2.f.b 4 60.l odd 4 1
2600.2.f.b 4 780.w odd 4 1
2600.2.k.a 4 60.h even 2 1
2600.2.k.a 4 780.d even 2 1
2704.2.a.s 2 39.f even 4 1
2704.2.a.t 2 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{4} + 9T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} + 21T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 36T_{11}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$7$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$37$ \( T^{4} + 121T^{2} + 2704 \) Copy content Toggle raw display
$41$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 15 T + 52)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 69T^{2} + 676 \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T - 52)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 14 T + 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 84T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{4} + 77T^{2} + 1444 \) Copy content Toggle raw display
$73$ \( T^{4} + 324 T^{2} + 20736 \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 68)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
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