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Properties

Label 2800.2.g.u
Level $2800$
Weight $2$
Character orbit 2800.g
Analytic conductor $22.358$
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] = [2800,2,Mod(449,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.g (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: \(22.3581125660\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 280)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{7} + (\beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{2} q^{7} + (\beta_{3} - 2) q^{9} + ( - \beta_{3} + 1) q^{11} + (2 \beta_{2} + 3 \beta_1) q^{13} + ( - 6 \beta_{2} - \beta_1) q^{17} + ( - 2 \beta_{3} - 2) q^{19} + ( - \beta_{3} + 1) q^{21} - 2 \beta_1 q^{23} + (4 \beta_{2} + \beta_1) q^{27} + (\beta_{3} - 3) q^{29} + ( - 4 \beta_{3} + 4) q^{31} + ( - 4 \beta_{2} + \beta_1) q^{33} + (2 \beta_{2} + 4 \beta_1) q^{37} + (\beta_{3} - 13) q^{39} + ( - 2 \beta_{3} + 4) q^{41} + (4 \beta_{2} + 2 \beta_1) q^{43} + (4 \beta_{2} - \beta_1) q^{47} - q^{49} + (5 \beta_{3} - 1) q^{51} + ( - 10 \beta_{2} - 2 \beta_1) q^{53} + ( - 8 \beta_{2} - 2 \beta_1) q^{57} - 4 q^{59} + (2 \beta_{3} - 12) q^{61} + ( - \beta_{2} + \beta_1) q^{63} + ( - 4 \beta_{2} + 4 \beta_1) q^{67} + ( - 2 \beta_{3} + 10) q^{69} + (2 \beta_{2} - 4 \beta_1) q^{73} - \beta_1 q^{77} + (3 \beta_{3} - 7) q^{79} - 7 q^{81} - 12 \beta_{2} q^{83} + (4 \beta_{2} - 3 \beta_1) q^{87} - 2 \beta_{3} q^{89} + ( - 3 \beta_{3} + 1) q^{91} + ( - 16 \beta_{2} + 4 \beta_1) q^{93} + ( - 6 \beta_{2} + 3 \beta_1) q^{97} + (2 \beta_{3} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{9} + 2 q^{11} - 12 q^{19} + 2 q^{21} - 10 q^{29} + 8 q^{31} - 50 q^{39} + 12 q^{41} - 4 q^{49} + 6 q^{51} - 16 q^{59} - 44 q^{61} + 36 q^{69} - 22 q^{79} - 28 q^{81} - 4 q^{89} - 2 q^{91} - 20 q^{99}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) 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
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
− 2.56155i
− 1.56155i
1.56155i
2.56155i
0 − 2.56155i 0 0 0 1.00000i 0 −3.56155 0
449.2 0 − 1.56155i 0 0 0 − 1.00000i 0 0.561553 0
449.3 0 1.56155i 0 0 0 1.00000i 0 0.561553 0
449.4 0 2.56155i 0 0 0 − 1.00000i 0 −3.56155 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.g.u 4
4.b odd 2 1 1400.2.g.k 4
5.b even 2 1 inner 2800.2.g.u 4
5.c odd 4 1 560.2.a.g 2
5.c odd 4 1 2800.2.a.bn 2
15.e even 4 1 5040.2.a.bq 2
20.d odd 2 1 1400.2.g.k 4
20.e even 4 1 280.2.a.d ✓ 2
20.e even 4 1 1400.2.a.p 2
35.f even 4 1 3920.2.a.bu 2
40.i odd 4 1 2240.2.a.bi 2
40.k even 4 1 2240.2.a.be 2
60.l odd 4 1 2520.2.a.w 2
140.j odd 4 1 1960.2.a.r 2
140.j odd 4 1 9800.2.a.by 2
140.w even 12 2 1960.2.q.s 4
140.x odd 12 2 1960.2.q.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.d ✓ 2 20.e even 4 1
560.2.a.g 2 5.c odd 4 1
1400.2.a.p 2 20.e even 4 1
1400.2.g.k 4 4.b odd 2 1
1400.2.g.k 4 20.d odd 2 1
1960.2.a.r 2 140.j odd 4 1
1960.2.q.s 4 140.w even 12 2
1960.2.q.u 4 140.x odd 12 2
2240.2.a.be 2 40.k even 4 1
2240.2.a.bi 2 40.i odd 4 1
2520.2.a.w 2 60.l odd 4 1
2800.2.a.bn 2 5.c odd 4 1
2800.2.g.u 4 1.a even 1 1 trivial
2800.2.g.u 4 5.b even 2 1 inner
3920.2.a.bu 2 35.f even 4 1
5040.2.a.bq 2 15.e even 4 1
9800.2.a.by 2 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2800, [\chi])\):

\( T_{3}^{4} + 9T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{4} + 77T_{13}^{2} + 1444 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 77T^{2} + 1444 \) Copy content Toggle raw display
$17$ \( T^{4} + 69T^{2} + 676 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 68)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 52T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 49T^{2} + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 196T^{2} + 4096 \) Copy content Toggle raw display
$59$ \( (T + 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 22 T + 104)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 208T^{2} + 1024 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 2704 \) Copy content Toggle raw display
$79$ \( (T^{2} + 11 T - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 189T^{2} + 324 \) Copy content Toggle raw display
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