Properties

Label 400.2.c.b
Level $400$
Weight $2$
Character orbit 400.c
Analytic conductor $3.194$
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] = [400,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(3.19401608085\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{7} - q^{9} + \beta q^{13} + 3 \beta q^{17} - 4 q^{19} - 4 q^{21} - 3 \beta q^{23} + 2 \beta q^{27} - 6 q^{29} + 4 q^{31} - \beta q^{37} - 4 q^{39} + 6 q^{41} + 5 \beta q^{43} - 3 \beta q^{47} + 3 q^{49} - 12 q^{51} - 3 \beta q^{53} - 4 \beta q^{57} + 12 q^{59} + 2 q^{61} - \beta q^{63} + \beta q^{67} + 12 q^{69} + 12 q^{71} + \beta q^{73} + 8 q^{79} - 11 q^{81} - 3 \beta q^{83} - 6 \beta q^{87} + 6 q^{89} - 4 q^{91} + 4 \beta q^{93} - \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} - 8 q^{19} - 8 q^{21} - 12 q^{29} + 8 q^{31} - 8 q^{39} + 12 q^{41} + 6 q^{49} - 24 q^{51} + 24 q^{59} + 4 q^{61} + 24 q^{69} + 24 q^{71} + 16 q^{79} - 22 q^{81} + 12 q^{89} - 8 q^{91}+O(q^{100}) \) 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 2.00000i 0 0 0 2.00000i 0 −1.00000 0
49.2 0 2.00000i 0 0 0 2.00000i 0 −1.00000 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 400.2.c.b 2
3.b odd 2 1 3600.2.f.j 2
4.b odd 2 1 100.2.c.a 2
5.b even 2 1 inner 400.2.c.b 2
5.c odd 4 1 80.2.a.b 1
5.c odd 4 1 400.2.a.c 1
8.b even 2 1 1600.2.c.e 2
8.d odd 2 1 1600.2.c.d 2
12.b even 2 1 900.2.d.c 2
15.d odd 2 1 3600.2.f.j 2
15.e even 4 1 720.2.a.h 1
15.e even 4 1 3600.2.a.be 1
20.d odd 2 1 100.2.c.a 2
20.e even 4 1 20.2.a.a 1
20.e even 4 1 100.2.a.a 1
28.d even 2 1 4900.2.e.f 2
35.f even 4 1 3920.2.a.h 1
40.e odd 2 1 1600.2.c.d 2
40.f even 2 1 1600.2.c.e 2
40.i odd 4 1 320.2.a.a 1
40.i odd 4 1 1600.2.a.w 1
40.k even 4 1 320.2.a.f 1
40.k even 4 1 1600.2.a.c 1
55.e even 4 1 9680.2.a.ba 1
60.h even 2 1 900.2.d.c 2
60.l odd 4 1 180.2.a.a 1
60.l odd 4 1 900.2.a.b 1
80.i odd 4 1 1280.2.d.g 2
80.j even 4 1 1280.2.d.c 2
80.s even 4 1 1280.2.d.c 2
80.t odd 4 1 1280.2.d.g 2
120.q odd 4 1 2880.2.a.m 1
120.w even 4 1 2880.2.a.f 1
140.c even 2 1 4900.2.e.f 2
140.j odd 4 1 980.2.a.h 1
140.j odd 4 1 4900.2.a.e 1
140.w even 12 2 980.2.i.i 2
140.x odd 12 2 980.2.i.c 2
180.v odd 12 2 1620.2.i.b 2
180.x even 12 2 1620.2.i.h 2
220.i odd 4 1 2420.2.a.a 1
260.l odd 4 1 3380.2.f.b 2
260.p even 4 1 3380.2.a.c 1
260.s odd 4 1 3380.2.f.b 2
340.i even 4 1 5780.2.c.a 2
340.r even 4 1 5780.2.a.f 1
340.s even 4 1 5780.2.c.a 2
380.j odd 4 1 7220.2.a.f 1
420.w even 4 1 8820.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 20.e even 4 1
80.2.a.b 1 5.c odd 4 1
100.2.a.a 1 20.e even 4 1
100.2.c.a 2 4.b odd 2 1
100.2.c.a 2 20.d odd 2 1
180.2.a.a 1 60.l odd 4 1
320.2.a.a 1 40.i odd 4 1
320.2.a.f 1 40.k even 4 1
400.2.a.c 1 5.c odd 4 1
400.2.c.b 2 1.a even 1 1 trivial
400.2.c.b 2 5.b even 2 1 inner
720.2.a.h 1 15.e even 4 1
900.2.a.b 1 60.l odd 4 1
900.2.d.c 2 12.b even 2 1
900.2.d.c 2 60.h even 2 1
980.2.a.h 1 140.j odd 4 1
980.2.i.c 2 140.x odd 12 2
980.2.i.i 2 140.w even 12 2
1280.2.d.c 2 80.j even 4 1
1280.2.d.c 2 80.s even 4 1
1280.2.d.g 2 80.i odd 4 1
1280.2.d.g 2 80.t odd 4 1
1600.2.a.c 1 40.k even 4 1
1600.2.a.w 1 40.i odd 4 1
1600.2.c.d 2 8.d odd 2 1
1600.2.c.d 2 40.e odd 2 1
1600.2.c.e 2 8.b even 2 1
1600.2.c.e 2 40.f even 2 1
1620.2.i.b 2 180.v odd 12 2
1620.2.i.h 2 180.x even 12 2
2420.2.a.a 1 220.i odd 4 1
2880.2.a.f 1 120.w even 4 1
2880.2.a.m 1 120.q odd 4 1
3380.2.a.c 1 260.p even 4 1
3380.2.f.b 2 260.l odd 4 1
3380.2.f.b 2 260.s odd 4 1
3600.2.a.be 1 15.e even 4 1
3600.2.f.j 2 3.b odd 2 1
3600.2.f.j 2 15.d odd 2 1
3920.2.a.h 1 35.f even 4 1
4900.2.a.e 1 140.j odd 4 1
4900.2.e.f 2 28.d even 2 1
4900.2.e.f 2 140.c even 2 1
5780.2.a.f 1 340.r even 4 1
5780.2.c.a 2 340.i even 4 1
5780.2.c.a 2 340.s even 4 1
7220.2.a.f 1 380.j odd 4 1
8820.2.a.g 1 420.w even 4 1
9680.2.a.ba 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 100 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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