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Properties

Label 2800.2.k.l
Level $2800$
Weight $2$
Character orbit 2800.k
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(2351,2800)]
 
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, 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("2800.2351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.k (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: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{3} - \beta_{2} - \beta_1) q^{7} - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{3} - \beta_{2} - \beta_1) q^{7} - \beta_{6} q^{9} - \beta_1 q^{11} + (\beta_{5} - \beta_{4}) q^{13} - \beta_{4} q^{17} + ( - \beta_{7} - \beta_{2}) q^{19} + ( - \beta_{6} - \beta_{5} + 3) q^{21} + ( - 2 \beta_{3} - 2 \beta_1) q^{23} - \beta_{7} q^{27} + ( - \beta_{6} - 3) q^{29} - 2 \beta_{7} q^{31} + ( - 2 \beta_{5} - \beta_{4}) q^{33} + (2 \beta_{6} - 4) q^{37} + ( - 2 \beta_{3} + \beta_1) q^{39} + (2 \beta_{5} + 2 \beta_{4}) q^{41} - 2 \beta_{3} q^{43} + (3 \beta_{7} + 2 \beta_{2}) q^{47} + ( - 2 \beta_{6} - 2 \beta_{5} - 1) q^{49} + ( - 2 \beta_{3} - \beta_1) q^{51} + ( - 4 \beta_{6} - 2) q^{53} + ( - 2 \beta_{6} + 2) q^{57} + (\beta_{7} - 3 \beta_{2}) q^{59} + (3 \beta_{5} + 2 \beta_{4}) q^{61} + ( - \beta_{7} - 3 \beta_{3} + \cdots + \beta_1) q^{63}+ \cdots + ( - 2 \beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} + 28 q^{21} - 20 q^{29} - 40 q^{37} + 24 q^{57} - 16 q^{77} - 16 q^{81} - 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} + 20\nu^{2} + 27 ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 16\nu^{3} + 18\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{6} + 16\nu^{4} + 80\nu^{2} + 153 ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 16\nu^{3} + 72\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - 16\nu^{5} - 8\nu^{3} - 81\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 5\nu^{4} + 7\nu^{2} + 18 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - \nu^{5} - 14\nu^{3} - 27\nu ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 2\beta_{3} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 4\beta_{5} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{6} - 6\beta_{3} + 5\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8\beta_{7} - 8\beta_{5} - \beta_{4} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{3} - 16\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -24\beta_{7} - 24\beta_{5} - 13\beta_{4} - 11\beta_{2} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2351.1
−1.26217 − 1.18614i
−1.26217 + 1.18614i
−0.396143 + 1.68614i
−0.396143 − 1.68614i
0.396143 − 1.68614i
0.396143 + 1.68614i
1.26217 + 1.18614i
1.26217 − 1.18614i
0 −2.52434 0 0 0 −2.52434 − 0.792287i 0 3.37228 0
2351.2 0 −2.52434 0 0 0 −2.52434 + 0.792287i 0 3.37228 0
2351.3 0 −0.792287 0 0 0 −0.792287 − 2.52434i 0 −2.37228 0
2351.4 0 −0.792287 0 0 0 −0.792287 + 2.52434i 0 −2.37228 0
2351.5 0 0.792287 0 0 0 0.792287 − 2.52434i 0 −2.37228 0
2351.6 0 0.792287 0 0 0 0.792287 + 2.52434i 0 −2.37228 0
2351.7 0 2.52434 0 0 0 2.52434 − 0.792287i 0 3.37228 0
2351.8 0 2.52434 0 0 0 2.52434 + 0.792287i 0 3.37228 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2351.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{3}^{4} - 7T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 7T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{19}^{2} - 12 \) Copy content Toggle raw display
\( T_{37}^{2} + 10T_{37} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 34T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 29 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 17 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T - 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T - 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 84 T^{2} + 576)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 187 T^{2} + 7744)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 132)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 44)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 116 T^{2} + 64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 44)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 44)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 84 T^{2} + 576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 447 T^{2} + 49284)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 336 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 233 T^{2} + 1024)^{2} \) Copy content Toggle raw display
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