Properties

Label 400.3.p.f
Level $400$
Weight $3$
Character orbit 400.p
Analytic conductor $10.899$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,3,Mod(193,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.p (of order \(4\), degree \(2\), 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: \(10.8992105744\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{3} + (7 i + 7) q^{7} + 7 i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{3} + (7 i + 7) q^{7} + 7 i q^{9} + 4 q^{11} + (12 i - 12) q^{13} + ( - 20 i - 20) q^{17} + 28 i q^{19} + 14 q^{21} + ( - 9 i + 9) q^{23} + (16 i + 16) q^{27} + 34 i q^{29} + 40 q^{31} + ( - 4 i + 4) q^{33} + (16 i + 16) q^{37} + 24 i q^{39} + 32 q^{41} + ( - 7 i + 7) q^{43} + ( - 31 i - 31) q^{47} + 49 i q^{49} - 40 q^{51} + ( - 52 i + 52) q^{53} + (28 i + 28) q^{57} + 44 i q^{59} + (49 i - 49) q^{63} + ( - 81 i - 81) q^{67} - 18 i q^{69} + 112 q^{71} + (44 i - 44) q^{73} + (28 i + 28) q^{77} - 72 i q^{79} - 31 q^{81} + (49 i - 49) q^{83} + (34 i + 34) q^{87} - 98 i q^{89} - 168 q^{91} + ( - 40 i + 40) q^{93} + (44 i + 44) q^{97} + 28 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 14 q^{7} + 8 q^{11} - 24 q^{13} - 40 q^{17} + 28 q^{21} + 18 q^{23} + 32 q^{27} + 80 q^{31} + 8 q^{33} + 32 q^{37} + 64 q^{41} + 14 q^{43} - 62 q^{47} - 80 q^{51} + 104 q^{53} + 56 q^{57} - 98 q^{63} - 162 q^{67} + 224 q^{71} - 88 q^{73} + 56 q^{77} - 62 q^{81} - 98 q^{83} + 68 q^{87} - 336 q^{91} + 80 q^{93} + 88 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
1.00000i
1.00000i
0 1.00000 + 1.00000i 0 0 0 7.00000 7.00000i 0 7.00000i 0
257.1 0 1.00000 1.00000i 0 0 0 7.00000 + 7.00000i 0 7.00000i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.p.f 2
4.b odd 2 1 200.3.l.a 2
5.b even 2 1 400.3.p.c 2
5.c odd 4 1 400.3.p.c 2
5.c odd 4 1 inner 400.3.p.f 2
12.b even 2 1 1800.3.v.a 2
20.d odd 2 1 200.3.l.c yes 2
20.e even 4 1 200.3.l.a 2
20.e even 4 1 200.3.l.c yes 2
60.h even 2 1 1800.3.v.g 2
60.l odd 4 1 1800.3.v.a 2
60.l odd 4 1 1800.3.v.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.l.a 2 4.b odd 2 1
200.3.l.a 2 20.e even 4 1
200.3.l.c yes 2 20.d odd 2 1
200.3.l.c yes 2 20.e even 4 1
400.3.p.c 2 5.b even 2 1
400.3.p.c 2 5.c odd 4 1
400.3.p.f 2 1.a even 1 1 trivial
400.3.p.f 2 5.c odd 4 1 inner
1800.3.v.a 2 12.b even 2 1
1800.3.v.a 2 60.l odd 4 1
1800.3.v.g 2 60.h even 2 1
1800.3.v.g 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{2} - 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 14T_{7} + 98 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 24T + 288 \) Copy content Toggle raw display
$17$ \( T^{2} + 40T + 800 \) Copy content Toggle raw display
$19$ \( T^{2} + 784 \) Copy content Toggle raw display
$23$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$29$ \( T^{2} + 1156 \) Copy content Toggle raw display
$31$ \( (T - 40)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 32T + 512 \) Copy content Toggle raw display
$41$ \( (T - 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$47$ \( T^{2} + 62T + 1922 \) Copy content Toggle raw display
$53$ \( T^{2} - 104T + 5408 \) Copy content Toggle raw display
$59$ \( T^{2} + 1936 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 162T + 13122 \) Copy content Toggle raw display
$71$ \( (T - 112)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 88T + 3872 \) Copy content Toggle raw display
$79$ \( T^{2} + 5184 \) Copy content Toggle raw display
$83$ \( T^{2} + 98T + 4802 \) Copy content Toggle raw display
$89$ \( T^{2} + 9604 \) Copy content Toggle raw display
$97$ \( T^{2} - 88T + 3872 \) Copy content Toggle raw display
show more
show less