Properties

Label 400.7.h.b
Level $400$
Weight $7$
Character orbit 400.h
Analytic conductor $92.022$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,7,Mod(399,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.399");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 400.h (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: \(92.0216334479\)
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: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 16)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + 22 \beta_{2} q^{7} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + 22 \beta_{2} q^{7} + 39 q^{9} + 33 \beta_{3} q^{11} - 77 \beta_1 q^{13} + 3729 \beta_1 q^{17} + 77 \beta_{3} q^{19} - 16896 q^{21} - 306 \beta_{2} q^{23} + 690 \beta_{2} q^{27} + 10758 q^{29} + 72 \beta_{3} q^{31} - 12672 \beta_1 q^{33} - 5675 \beta_1 q^{37} + 154 \beta_{3} q^{39} + 67122 q^{41} - 2871 \beta_{2} q^{43} - 2508 \beta_{2} q^{47} + 254063 q^{49} - 7458 \beta_{3} q^{51} - 54981 \beta_1 q^{53} - 29568 \beta_1 q^{57} - 11037 \beta_{3} q^{59} + 306746 q^{61} + 858 \beta_{2} q^{63} - 7951 \beta_{2} q^{67} + 235008 q^{69} + 13434 \beta_{3} q^{71} - 82841 \beta_1 q^{73} + 278784 \beta_1 q^{77} + 27500 \beta_{3} q^{79} - 558351 q^{81} - 17853 \beta_{2} q^{83} - 10758 \beta_{2} q^{87} - 471954 q^{89} - 3388 \beta_{3} q^{91} - 27648 \beta_1 q^{93} + 455297 \beta_1 q^{97} + 1287 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 156 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 156 q^{9} - 67584 q^{21} + 43032 q^{29} + 268488 q^{41} + 1016252 q^{49} + 1226984 q^{61} + 940032 q^{69} - 2233404 q^{81} - 1887816 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -16\zeta_{12}^{3} + 32\zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 32\zeta_{12}^{2} - 16 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{2} + 8\beta_1 ) / 32 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{3} + 16 ) / 32 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
399.1
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 −27.7128 0 0 0 609.682 0 39.0000 0
399.2 0 −27.7128 0 0 0 609.682 0 39.0000 0
399.3 0 27.7128 0 0 0 −609.682 0 39.0000 0
399.4 0 27.7128 0 0 0 −609.682 0 39.0000 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.7.h.b 4
4.b odd 2 1 inner 400.7.h.b 4
5.b even 2 1 inner 400.7.h.b 4
5.c odd 4 1 16.7.c.b 2
5.c odd 4 1 400.7.b.c 2
15.e even 4 1 144.7.g.f 2
20.d odd 2 1 inner 400.7.h.b 4
20.e even 4 1 16.7.c.b 2
20.e even 4 1 400.7.b.c 2
40.i odd 4 1 64.7.c.d 2
40.k even 4 1 64.7.c.d 2
60.l odd 4 1 144.7.g.f 2
80.i odd 4 1 256.7.d.e 4
80.j even 4 1 256.7.d.e 4
80.s even 4 1 256.7.d.e 4
80.t odd 4 1 256.7.d.e 4
120.q odd 4 1 576.7.g.d 2
120.w even 4 1 576.7.g.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.7.c.b 2 5.c odd 4 1
16.7.c.b 2 20.e even 4 1
64.7.c.d 2 40.i odd 4 1
64.7.c.d 2 40.k even 4 1
144.7.g.f 2 15.e even 4 1
144.7.g.f 2 60.l odd 4 1
256.7.d.e 4 80.i odd 4 1
256.7.d.e 4 80.j even 4 1
256.7.d.e 4 80.s even 4 1
256.7.d.e 4 80.t odd 4 1
400.7.b.c 2 5.c odd 4 1
400.7.b.c 2 20.e even 4 1
400.7.h.b 4 1.a even 1 1 trivial
400.7.h.b 4 4.b odd 2 1 inner
400.7.h.b 4 5.b even 2 1 inner
400.7.h.b 4 20.d odd 2 1 inner
576.7.g.d 2 120.q odd 4 1
576.7.g.d 2 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 768 \) acting on \(S_{7}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 768)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 371712)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 836352)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 23716)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 55621764)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4553472)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 71912448)^{2} \) Copy content Toggle raw display
$29$ \( (T - 10758)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3981312)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 128822500)^{2} \) Copy content Toggle raw display
$41$ \( (T - 67122)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6330348288)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 4830769152)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12091641444)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 93554203392)^{2} \) Copy content Toggle raw display
$61$ \( (T - 306746)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 48551731968)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 138602769408)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 27450525124)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 580800000000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 244784339712)^{2} \) Copy content Toggle raw display
$89$ \( (T + 471954)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 829181432836)^{2} \) Copy content Toggle raw display
show more
show less