Properties

Label 400.4.a.w
Level $400$
Weight $4$
Character orbit 400.a
Self dual yes
Analytic conductor $23.601$
Analytic rank $0$
Dimension $2$
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] = [400,4,Mod(1,400)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

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: yes
Analytic conductor: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 2\sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{7} + 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{7} + 49 q^{9} - 20 q^{11} + 6 \beta q^{13} + 8 \beta q^{17} - 84 q^{19} + 76 q^{21} + 7 \beta q^{23} + 22 \beta q^{27} - 6 q^{29} + 224 q^{31} - 20 \beta q^{33} + 14 \beta q^{37} + 456 q^{39} + 266 q^{41} - 35 \beta q^{43} - 43 \beta q^{47} - 267 q^{49} + 608 q^{51} + 42 \beta q^{53} - 84 \beta q^{57} - 28 q^{59} + 182 q^{61} + 49 \beta q^{63} - 49 \beta q^{67} + 532 q^{69} - 408 q^{71} - 124 \beta q^{73} - 20 \beta q^{77} + 48 q^{79} + 349 q^{81} - 23 \beta q^{83} - 6 \beta q^{87} + 1526 q^{89} + 456 q^{91} + 224 \beta q^{93} + 64 \beta q^{97} - 980 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 98 q^{9} - 40 q^{11} - 168 q^{19} + 152 q^{21} - 12 q^{29} + 448 q^{31} + 912 q^{39} + 532 q^{41} - 534 q^{49} + 1216 q^{51} - 56 q^{59} + 364 q^{61} + 1064 q^{69} - 816 q^{71} + 96 q^{79} + 698 q^{81} + 3052 q^{89} + 912 q^{91} - 1960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.35890
4.35890
0 −8.71780 0 0 0 −8.71780 0 49.0000 0
1.2 0 8.71780 0 0 0 8.71780 0 49.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

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.a.w 2
4.b odd 2 1 100.4.a.d 2
5.b even 2 1 inner 400.4.a.w 2
5.c odd 4 2 80.4.c.b 2
8.b even 2 1 1600.4.a.ck 2
8.d odd 2 1 1600.4.a.cj 2
12.b even 2 1 900.4.a.s 2
15.e even 4 2 720.4.f.a 2
20.d odd 2 1 100.4.a.d 2
20.e even 4 2 20.4.c.a 2
40.e odd 2 1 1600.4.a.cj 2
40.f even 2 1 1600.4.a.ck 2
40.i odd 4 2 320.4.c.b 2
40.k even 4 2 320.4.c.a 2
60.h even 2 1 900.4.a.s 2
60.l odd 4 2 180.4.d.a 2
140.j odd 4 2 980.4.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.c.a 2 20.e even 4 2
80.4.c.b 2 5.c odd 4 2
100.4.a.d 2 4.b odd 2 1
100.4.a.d 2 20.d odd 2 1
180.4.d.a 2 60.l odd 4 2
320.4.c.a 2 40.k even 4 2
320.4.c.b 2 40.i odd 4 2
400.4.a.w 2 1.a even 1 1 trivial
400.4.a.w 2 5.b even 2 1 inner
720.4.f.a 2 15.e even 4 2
900.4.a.s 2 12.b even 2 1
900.4.a.s 2 60.h even 2 1
980.4.e.a 2 140.j odd 4 2
1600.4.a.cj 2 8.d odd 2 1
1600.4.a.cj 2 40.e odd 2 1
1600.4.a.ck 2 8.b even 2 1
1600.4.a.ck 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(400))\):

\( T_{3}^{2} - 76 \) Copy content Toggle raw display
\( T_{7}^{2} - 76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 76 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 76 \) Copy content Toggle raw display
$11$ \( (T + 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2736 \) Copy content Toggle raw display
$17$ \( T^{2} - 4864 \) Copy content Toggle raw display
$19$ \( (T + 84)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3724 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 14896 \) Copy content Toggle raw display
$41$ \( (T - 266)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 93100 \) Copy content Toggle raw display
$47$ \( T^{2} - 140524 \) Copy content Toggle raw display
$53$ \( T^{2} - 134064 \) Copy content Toggle raw display
$59$ \( (T + 28)^{2} \) Copy content Toggle raw display
$61$ \( (T - 182)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 182476 \) Copy content Toggle raw display
$71$ \( (T + 408)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1168576 \) Copy content Toggle raw display
$79$ \( (T - 48)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 40204 \) Copy content Toggle raw display
$89$ \( (T - 1526)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 311296 \) Copy content Toggle raw display
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