Properties

Label 400.6.c.i
Level $400$
Weight $6$
Character orbit 400.c
Analytic conductor $64.154$
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] = [400,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(64.1535279252\)
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 10)
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.

\(f(q)\) \(=\) \( q + 3 \beta q^{3} + 59 \beta q^{7} + 207 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta q^{3} + 59 \beta q^{7} + 207 q^{9} - 192 q^{11} - 553 \beta q^{13} + 381 \beta q^{17} - 2740 q^{19} - 708 q^{21} + 783 \beta q^{23} + 1350 \beta q^{27} - 5910 q^{29} + 6868 q^{31} - 576 \beta q^{33} - 2759 \beta q^{37} + 6636 q^{39} - 378 q^{41} - 1217 \beta q^{43} - 6561 \beta q^{47} + 2883 q^{49} - 4572 q^{51} + 4587 \beta q^{53} - 8220 \beta q^{57} - 34980 q^{59} - 9838 q^{61} + 12213 \beta q^{63} - 16861 \beta q^{67} - 9396 q^{69} - 70212 q^{71} - 10993 \beta q^{73} - 11328 \beta q^{77} + 4520 q^{79} + 34101 q^{81} - 54537 \beta q^{83} - 17730 \beta q^{87} - 38490 q^{89} + 130508 q^{91} + 20604 \beta q^{93} - 959 \beta q^{97} - 39744 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 414 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 414 q^{9} - 384 q^{11} - 5480 q^{19} - 1416 q^{21} - 11820 q^{29} + 13736 q^{31} + 13272 q^{39} - 756 q^{41} + 5766 q^{49} - 9144 q^{51} - 69960 q^{59} - 19676 q^{61} - 18792 q^{69} - 140424 q^{71} + 9040 q^{79} + 68202 q^{81} - 76980 q^{89} + 261016 q^{91} - 79488 q^{99}+O(q^{100}) \) 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 0 118.000i 0 207.000 0
49.2 0 6.00000i 0 0 0 118.000i 0 207.000 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 400.6.c.i 2
4.b odd 2 1 50.6.b.b 2
5.b even 2 1 inner 400.6.c.i 2
5.c odd 4 1 80.6.a.c 1
5.c odd 4 1 400.6.a.i 1
12.b even 2 1 450.6.c.f 2
15.e even 4 1 720.6.a.v 1
20.d odd 2 1 50.6.b.b 2
20.e even 4 1 10.6.a.c 1
20.e even 4 1 50.6.a.b 1
40.i odd 4 1 320.6.a.k 1
40.k even 4 1 320.6.a.f 1
60.h even 2 1 450.6.c.f 2
60.l odd 4 1 90.6.a.b 1
60.l odd 4 1 450.6.a.u 1
140.j odd 4 1 490.6.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 20.e even 4 1
50.6.a.b 1 20.e even 4 1
50.6.b.b 2 4.b odd 2 1
50.6.b.b 2 20.d odd 2 1
80.6.a.c 1 5.c odd 4 1
90.6.a.b 1 60.l odd 4 1
320.6.a.f 1 40.k even 4 1
320.6.a.k 1 40.i odd 4 1
400.6.a.i 1 5.c odd 4 1
400.6.c.i 2 1.a even 1 1 trivial
400.6.c.i 2 5.b even 2 1 inner
450.6.a.u 1 60.l odd 4 1
450.6.c.f 2 12.b even 2 1
450.6.c.f 2 60.h even 2 1
490.6.a.k 1 140.j odd 4 1
720.6.a.v 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 36 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 36 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 13924 \) Copy content Toggle raw display
$11$ \( (T + 192)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1223236 \) Copy content Toggle raw display
$17$ \( T^{2} + 580644 \) Copy content Toggle raw display
$19$ \( (T + 2740)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2452356 \) Copy content Toggle raw display
$29$ \( (T + 5910)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6868)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 30448324 \) Copy content Toggle raw display
$41$ \( (T + 378)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 5924356 \) Copy content Toggle raw display
$47$ \( T^{2} + 172186884 \) Copy content Toggle raw display
$53$ \( T^{2} + 84162276 \) Copy content Toggle raw display
$59$ \( (T + 34980)^{2} \) Copy content Toggle raw display
$61$ \( (T + 9838)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1137173284 \) Copy content Toggle raw display
$71$ \( (T + 70212)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 483384196 \) Copy content Toggle raw display
$79$ \( (T - 4520)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11897137476 \) Copy content Toggle raw display
$89$ \( (T + 38490)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 3678724 \) Copy content Toggle raw display
show more
show less