Properties

Label 400.2.c.c
Level $400$
Weight $2$
Character orbit 400.c
Analytic conductor $3.194$
Analytic rank $0$
Dimension $2$
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: \( 1 \)
Twist minimal: no (minimal twist has level 50)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} - 2 i q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - 2 i q^{7} + 2 q^{9} + 3 q^{11} + 4 i q^{13} - 3 i q^{17} + 5 q^{19} + 2 q^{21} + 6 i q^{23} + 5 i q^{27} - 2 q^{31} + 3 i q^{33} + 2 i q^{37} - 4 q^{39} - 3 q^{41} - 4 i q^{43} - 12 i q^{47} + 3 q^{49} + 3 q^{51} - 6 i q^{53} + 5 i q^{57} + 2 q^{61} - 4 i q^{63} + 13 i q^{67} - 6 q^{69} - 12 q^{71} - 11 i q^{73} - 6 i q^{77} - 10 q^{79} + q^{81} - 9 i q^{83} - 15 q^{89} + 8 q^{91} - 2 i q^{93} + 2 i q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} + 6 q^{11} + 10 q^{19} + 4 q^{21} - 4 q^{31} - 8 q^{39} - 6 q^{41} + 6 q^{49} + 6 q^{51} + 4 q^{61} - 12 q^{69} - 24 q^{71} - 20 q^{79} + 2 q^{81} - 30 q^{89} + 16 q^{91} + 12 q^{99}+O(q^{100}) \) 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 1.00000i 0 0 0 2.00000i 0 2.00000 0
49.2 0 1.00000i 0 0 0 2.00000i 0 2.00000 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.c 2
3.b odd 2 1 3600.2.f.f 2
4.b odd 2 1 50.2.b.a 2
5.b even 2 1 inner 400.2.c.c 2
5.c odd 4 1 400.2.a.d 1
5.c odd 4 1 400.2.a.f 1
8.b even 2 1 1600.2.c.h 2
8.d odd 2 1 1600.2.c.i 2
12.b even 2 1 450.2.c.c 2
15.d odd 2 1 3600.2.f.f 2
15.e even 4 1 3600.2.a.l 1
15.e even 4 1 3600.2.a.bc 1
20.d odd 2 1 50.2.b.a 2
20.e even 4 1 50.2.a.a 1
20.e even 4 1 50.2.a.b yes 1
28.d even 2 1 2450.2.c.m 2
40.e odd 2 1 1600.2.c.i 2
40.f even 2 1 1600.2.c.h 2
40.i odd 4 1 1600.2.a.i 1
40.i odd 4 1 1600.2.a.p 1
40.k even 4 1 1600.2.a.j 1
40.k even 4 1 1600.2.a.q 1
60.h even 2 1 450.2.c.c 2
60.l odd 4 1 450.2.a.c 1
60.l odd 4 1 450.2.a.g 1
140.c even 2 1 2450.2.c.m 2
140.j odd 4 1 2450.2.a.g 1
140.j odd 4 1 2450.2.a.bd 1
220.i odd 4 1 6050.2.a.h 1
220.i odd 4 1 6050.2.a.bi 1
260.p even 4 1 8450.2.a.d 1
260.p even 4 1 8450.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.a.a 1 20.e even 4 1
50.2.a.b yes 1 20.e even 4 1
50.2.b.a 2 4.b odd 2 1
50.2.b.a 2 20.d odd 2 1
400.2.a.d 1 5.c odd 4 1
400.2.a.f 1 5.c odd 4 1
400.2.c.c 2 1.a even 1 1 trivial
400.2.c.c 2 5.b even 2 1 inner
450.2.a.c 1 60.l odd 4 1
450.2.a.g 1 60.l odd 4 1
450.2.c.c 2 12.b even 2 1
450.2.c.c 2 60.h even 2 1
1600.2.a.i 1 40.i odd 4 1
1600.2.a.j 1 40.k even 4 1
1600.2.a.p 1 40.i odd 4 1
1600.2.a.q 1 40.k even 4 1
1600.2.c.h 2 8.b even 2 1
1600.2.c.h 2 40.f even 2 1
1600.2.c.i 2 8.d odd 2 1
1600.2.c.i 2 40.e odd 2 1
2450.2.a.g 1 140.j odd 4 1
2450.2.a.bd 1 140.j odd 4 1
2450.2.c.m 2 28.d even 2 1
2450.2.c.m 2 140.c even 2 1
3600.2.a.l 1 15.e even 4 1
3600.2.a.bc 1 15.e even 4 1
3600.2.f.f 2 3.b odd 2 1
3600.2.f.f 2 15.d odd 2 1
6050.2.a.h 1 220.i odd 4 1
6050.2.a.bi 1 220.i odd 4 1
8450.2.a.d 1 260.p even 4 1
8450.2.a.v 1 260.p even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1 \) 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
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 169 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 81 \) Copy content Toggle raw display
$89$ \( (T + 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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