Properties

Label 3100.3.f.a
Level $3100$
Weight $3$
Character orbit 3100.f
Analytic conductor $84.469$
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] = [3100,3,Mod(1549,3100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3100.1549");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3100.f (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: \(84.4688819517\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 124)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{7} + 3 q^{9} - \beta_{3} q^{11} + 5 \beta_1 q^{13} - 4 \beta_1 q^{17} + 14 q^{19} - 2 \beta_{3} q^{21} + 10 \beta_1 q^{23} - 6 \beta_1 q^{27} - \beta_{3} q^{29} - 31 q^{31} - 6 \beta_{2} q^{33} + \beta_1 q^{37} + 60 q^{39} + 54 q^{41} - 21 \beta_1 q^{43} + 3 \beta_{2} q^{47} - 51 q^{49} - 48 q^{51} - 9 \beta_1 q^{53} + 14 \beta_1 q^{57} + 6 q^{59} + \beta_{3} q^{61} - 3 \beta_{2} q^{63} - 11 \beta_{2} q^{67} + 120 q^{69} + 66 q^{71} + 26 \beta_1 q^{73} - 50 \beta_1 q^{77} - 4 \beta_{3} q^{79} - 99 q^{81} - 11 \beta_1 q^{83} - 6 \beta_{2} q^{87} + 2 \beta_{3} q^{89} - 10 \beta_{3} q^{91} - 31 \beta_1 q^{93} + 11 \beta_{2} q^{97} - 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} + 56 q^{19} - 124 q^{31} + 240 q^{39} + 216 q^{41} - 204 q^{49} - 192 q^{51} + 24 q^{59} + 480 q^{69} + 264 q^{71} - 396 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 10\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 20\zeta_{12}^{2} - 10 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{2} + 5\beta_1 ) / 20 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{3} + 10 ) / 20 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{2} ) / 10 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times\).

\(n\) \(1551\) \(1801\) \(2977\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1549.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 −3.46410 0 0 0 10.0000i 0 3.00000 0
1549.2 0 −3.46410 0 0 0 10.0000i 0 3.00000 0
1549.3 0 3.46410 0 0 0 10.0000i 0 3.00000 0
1549.4 0 3.46410 0 0 0 10.0000i 0 3.00000 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
5.b even 2 1 inner
31.b odd 2 1 inner
155.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3100.3.f.a 4
5.b even 2 1 inner 3100.3.f.a 4
5.c odd 4 1 124.3.c.a 2
5.c odd 4 1 3100.3.d.a 2
15.e even 4 1 1116.3.h.c 2
20.e even 4 1 496.3.e.a 2
31.b odd 2 1 inner 3100.3.f.a 4
155.c odd 2 1 inner 3100.3.f.a 4
155.f even 4 1 124.3.c.a 2
155.f even 4 1 3100.3.d.a 2
465.m odd 4 1 1116.3.h.c 2
620.m odd 4 1 496.3.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.3.c.a 2 5.c odd 4 1
124.3.c.a 2 155.f even 4 1
496.3.e.a 2 20.e even 4 1
496.3.e.a 2 620.m odd 4 1
1116.3.h.c 2 15.e even 4 1
1116.3.h.c 2 465.m odd 4 1
3100.3.d.a 2 5.c odd 4 1
3100.3.d.a 2 155.f even 4 1
3100.3.f.a 4 1.a even 1 1 trivial
3100.3.f.a 4 5.b even 2 1 inner
3100.3.f.a 4 31.b odd 2 1 inner
3100.3.f.a 4 155.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 12 \) acting on \(S_{3}^{\mathrm{new}}(3100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$19$ \( (T - 14)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 1200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
$31$ \( (T + 31)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$41$ \( (T - 54)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 5292)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 900)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 972)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12100)^{2} \) Copy content Toggle raw display
$71$ \( (T - 66)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 8112)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4800)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1452)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 12100)^{2} \) Copy content Toggle raw display
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