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Properties

Label 2800.2.e.h
Level $2800$
Weight $2$
Character orbit 2800.e
Analytic conductor $22.358$
Analytic rank $0$
Dimension $8$
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] = [2800,2,Mod(2799,2800)]
 
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([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.2799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.e (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: \(8\)
Coefficient field: 8.0.116319195136.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 77x^{4} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 560)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} + \beta_{6} q^{7} + ( - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{3} + \beta_{6} q^{7} + ( - \beta_1 - 1) q^{9} + (\beta_{7} - \beta_{6} + \beta_{5}) q^{11} + \beta_1 q^{13} + \beta_1 q^{17} + ( - \beta_{6} - \beta_{5} + \beta_{2}) q^{19} + (\beta_{4} - 2 \beta_{3} - 1) q^{21} + ( - \beta_{6} - \beta_{5}) q^{23} + (3 \beta_{7} + \beta_{6} - \beta_{5}) q^{27} + \beta_1 q^{29} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{31} + (\beta_1 + 6) q^{33} + (2 \beta_{4} - 2 \beta_{3}) q^{37} + ( - 5 \beta_{7} - \beta_{6} + \beta_{5}) q^{39} - 3 \beta_{3} q^{41} + 3 \beta_{2} q^{43} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5}) q^{47} + (\beta_{4} + \beta_{3} + \beta_1 + 2) q^{49} + ( - 5 \beta_{7} - \beta_{6} + \beta_{5}) q^{51} + 3 \beta_{3} q^{53} + ( - 2 \beta_{4} + 3 \beta_{3}) q^{57} + (\beta_{6} + \beta_{5} + 3 \beta_{2}) q^{59} + (2 \beta_{4} + 2 \beta_{3}) q^{61} + (\beta_{7} - \beta_{6} + \cdots - 3 \beta_{2}) q^{63}+ \cdots + ( - 8 \beta_{7} - 4 \beta_{6} + 4 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{9} + 4 q^{13} + 4 q^{17} - 8 q^{21} + 4 q^{29} + 52 q^{33} + 20 q^{49} + 48 q^{73} + 44 q^{77} + 56 q^{81} + 100 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 77x^{4} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} + 43 ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 79\nu^{3} + 18\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 79\nu^{2} ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} - 149\nu^{2} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{7} + 2\nu^{5} - 535\nu^{3} + 158\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{7} + 2\nu^{5} - 693\nu^{3} + 122\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} + 2\nu^{5} + 693\nu^{3} + 158\nu ) / 36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 4\beta_{6} + 5\beta_{5} - 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta _1 - 43 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} + 44\beta_{6} - 35\beta_{5} - 35\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -79\beta_{4} - 149\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 79\beta_{7} + 307\beta_{6} - 386\beta_{5} + 307\beta_{2} \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2799.1
−0.337637 + 0.337637i
0.337637 + 0.337637i
2.09428 + 2.09428i
−2.09428 + 2.09428i
2.09428 − 2.09428i
−2.09428 − 2.09428i
−0.337637 − 0.337637i
0.337637 − 0.337637i
0 − 2.96176i 0 0 0 −2.62412 − 0.337637i 0 −5.77200 0
2799.2 0 − 2.96176i 0 0 0 2.62412 − 0.337637i 0 −5.77200 0
2799.3 0 − 0.477491i 0 0 0 −1.61679 − 2.09428i 0 2.77200 0
2799.4 0 − 0.477491i 0 0 0 1.61679 − 2.09428i 0 2.77200 0
2799.5 0 0.477491i 0 0 0 −1.61679 + 2.09428i 0 2.77200 0
2799.6 0 0.477491i 0 0 0 1.61679 + 2.09428i 0 2.77200 0
2799.7 0 2.96176i 0 0 0 −2.62412 + 0.337637i 0 −5.77200 0
2799.8 0 2.96176i 0 0 0 2.62412 + 0.337637i 0 −5.77200 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2799.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
35.c odd 2 1 inner
140.c 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.e.h 8
4.b odd 2 1 inner 2800.2.e.h 8
5.b even 2 1 2800.2.e.g 8
5.c odd 4 1 560.2.k.b ✓ 8
5.c odd 4 1 2800.2.k.m 8
7.b odd 2 1 2800.2.e.g 8
15.e even 4 1 5040.2.d.d 8
20.d odd 2 1 2800.2.e.g 8
20.e even 4 1 560.2.k.b ✓ 8
20.e even 4 1 2800.2.k.m 8
28.d even 2 1 2800.2.e.g 8
35.c odd 2 1 inner 2800.2.e.h 8
35.f even 4 1 560.2.k.b ✓ 8
35.f even 4 1 2800.2.k.m 8
40.i odd 4 1 2240.2.k.d 8
40.k even 4 1 2240.2.k.d 8
60.l odd 4 1 5040.2.d.d 8
105.k odd 4 1 5040.2.d.d 8
140.c even 2 1 inner 2800.2.e.h 8
140.j odd 4 1 560.2.k.b ✓ 8
140.j odd 4 1 2800.2.k.m 8
280.s even 4 1 2240.2.k.d 8
280.y odd 4 1 2240.2.k.d 8
420.w even 4 1 5040.2.d.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.k.b ✓ 8 5.c odd 4 1
560.2.k.b ✓ 8 20.e even 4 1
560.2.k.b ✓ 8 35.f even 4 1
560.2.k.b ✓ 8 140.j odd 4 1
2240.2.k.d 8 40.i odd 4 1
2240.2.k.d 8 40.k even 4 1
2240.2.k.d 8 280.s even 4 1
2240.2.k.d 8 280.y odd 4 1
2800.2.e.g 8 5.b even 2 1
2800.2.e.g 8 7.b odd 2 1
2800.2.e.g 8 20.d odd 2 1
2800.2.e.g 8 28.d even 2 1
2800.2.e.h 8 1.a even 1 1 trivial
2800.2.e.h 8 4.b odd 2 1 inner
2800.2.e.h 8 35.c odd 2 1 inner
2800.2.e.h 8 140.c even 2 1 inner
2800.2.k.m 8 5.c odd 4 1
2800.2.k.m 8 20.e even 4 1
2800.2.k.m 8 35.f even 4 1
2800.2.k.m 8 140.j odd 4 1
5040.2.d.d 8 15.e even 4 1
5040.2.d.d 8 60.l odd 4 1
5040.2.d.d 8 105.k odd 4 1
5040.2.d.d 8 420.w even 4 1

Hecke kernels

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

\( T_{3}^{4} + 9T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{4} + 35T_{11}^{2} + 288 \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} - 18 \) Copy content Toggle raw display
\( T_{19}^{4} - 76T_{19}^{2} + 1152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9 T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 10 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 35 T^{2} + 288)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - T - 18)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - T - 18)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 76 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 38 T^{2} + 288)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 18)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 36 T^{2} + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 164 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 162 T^{2} + 648)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 65 T^{2} + 162)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 140 T^{2} + 4608)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 196 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 194 T^{2} + 5832)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 140 T^{2} + 4608)^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 171 T^{2} + 5832)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 162 T^{2} + 648)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 25 T + 138)^{4} \) Copy content Toggle raw display
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