Properties

Label 400.4.c.f
Level $400$
Weight $4$
Character orbit 400.c
Analytic conductor $23.601$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(49,400)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (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: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta q^{3} - 17 \beta q^{7} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta q^{3} - 17 \beta q^{7} - 9 q^{9} - 16 q^{11} + 29 \beta q^{13} + 35 \beta q^{17} + 4 q^{19} + 204 q^{21} + 67 \beta q^{23} + 54 \beta q^{27} + 242 q^{29} - 100 q^{31} - 48 \beta q^{33} + 219 \beta q^{37} - 348 q^{39} - 138 q^{41} - 89 \beta q^{43} + 11 \beta q^{47} - 813 q^{49} - 420 q^{51} + 81 \beta q^{53} + 12 \beta q^{57} - 268 q^{59} + 250 q^{61} + 153 \beta q^{63} + 211 \beta q^{67} - 804 q^{69} + 852 q^{71} + 153 \beta q^{73} + 272 \beta q^{77} - 456 q^{79} - 891 q^{81} - 217 \beta q^{83} + 726 \beta q^{87} + 726 q^{89} + 1972 q^{91} - 300 \beta q^{93} - 689 \beta q^{97} + 144 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 32 q^{11} + 8 q^{19} + 408 q^{21} + 484 q^{29} - 200 q^{31} - 696 q^{39} - 276 q^{41} - 1626 q^{49} - 840 q^{51} - 536 q^{59} + 500 q^{61} - 1608 q^{69} + 1704 q^{71} - 912 q^{79} - 1782 q^{81} + 1452 q^{89} + 3944 q^{91} + 288 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 6.00000i 0 0 0 34.0000i 0 −9.00000 0
49.2 0 6.00000i 0 0 0 34.0000i 0 −9.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.4.c.f 2
4.b odd 2 1 200.4.c.c 2
5.b even 2 1 inner 400.4.c.f 2
5.c odd 4 1 80.4.a.e 1
5.c odd 4 1 400.4.a.e 1
12.b even 2 1 1800.4.f.j 2
15.e even 4 1 720.4.a.bd 1
20.d odd 2 1 200.4.c.c 2
20.e even 4 1 40.4.a.a 1
20.e even 4 1 200.4.a.i 1
40.i odd 4 1 320.4.a.c 1
40.i odd 4 1 1600.4.a.br 1
40.k even 4 1 320.4.a.l 1
40.k even 4 1 1600.4.a.j 1
60.h even 2 1 1800.4.f.j 2
60.l odd 4 1 360.4.a.h 1
60.l odd 4 1 1800.4.a.bi 1
80.i odd 4 1 1280.4.d.a 2
80.j even 4 1 1280.4.d.p 2
80.s even 4 1 1280.4.d.p 2
80.t odd 4 1 1280.4.d.a 2
140.j odd 4 1 1960.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.a 1 20.e even 4 1
80.4.a.e 1 5.c odd 4 1
200.4.a.i 1 20.e even 4 1
200.4.c.c 2 4.b odd 2 1
200.4.c.c 2 20.d odd 2 1
320.4.a.c 1 40.i odd 4 1
320.4.a.l 1 40.k even 4 1
360.4.a.h 1 60.l odd 4 1
400.4.a.e 1 5.c odd 4 1
400.4.c.f 2 1.a even 1 1 trivial
400.4.c.f 2 5.b even 2 1 inner
720.4.a.bd 1 15.e even 4 1
1280.4.d.a 2 80.i odd 4 1
1280.4.d.a 2 80.t odd 4 1
1280.4.d.p 2 80.j even 4 1
1280.4.d.p 2 80.s even 4 1
1600.4.a.j 1 40.k even 4 1
1600.4.a.br 1 40.i odd 4 1
1800.4.a.bi 1 60.l odd 4 1
1800.4.f.j 2 12.b even 2 1
1800.4.f.j 2 60.h even 2 1
1960.4.a.h 1 140.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 \(S_{4}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{2} + 1156 \) Copy content Toggle raw display
\( T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 36 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1156 \) Copy content Toggle raw display
$11$ \( (T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3364 \) Copy content Toggle raw display
$17$ \( T^{2} + 4900 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 17956 \) Copy content Toggle raw display
$29$ \( (T - 242)^{2} \) Copy content Toggle raw display
$31$ \( (T + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 191844 \) Copy content Toggle raw display
$41$ \( (T + 138)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 31684 \) Copy content Toggle raw display
$47$ \( T^{2} + 484 \) Copy content Toggle raw display
$53$ \( T^{2} + 26244 \) Copy content Toggle raw display
$59$ \( (T + 268)^{2} \) Copy content Toggle raw display
$61$ \( (T - 250)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 178084 \) Copy content Toggle raw display
$71$ \( (T - 852)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 93636 \) Copy content Toggle raw display
$79$ \( (T + 456)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 188356 \) Copy content Toggle raw display
$89$ \( (T - 726)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1898884 \) Copy content Toggle raw display
show more
show less