Properties

Label 400.6.a.t
Level $400$
Weight $6$
Character orbit 400.a
Self dual yes
Analytic conductor $64.154$
Analytic rank $1$
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] = [400,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(64.1535279252\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
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 \(\beta = 2\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta q^{3} - 9 \beta q^{7} + 153 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta q^{3} - 9 \beta q^{7} + 153 q^{9} - 252 q^{11} - 18 \beta q^{13} + 104 \beta q^{17} - 220 q^{19} - 1188 q^{21} - 367 \beta q^{23} - 270 \beta q^{27} + 6930 q^{29} - 6752 q^{31} - 756 \beta q^{33} - 2106 \beta q^{37} - 2376 q^{39} - 198 q^{41} + 63 \beta q^{43} - 1589 \beta q^{47} - 13243 q^{49} + 13728 q^{51} - 878 \beta q^{53} - 660 \beta q^{57} - 24660 q^{59} - 5698 q^{61} - 1377 \beta q^{63} - 6579 \beta q^{67} - 48444 q^{69} - 53352 q^{71} + 10692 \beta q^{73} + 2268 \beta q^{77} + 51920 q^{79} - 72819 q^{81} + 9323 \beta q^{83} + 20790 \beta q^{87} + 9990 q^{89} + 7128 q^{91} - 20256 \beta q^{93} + 15264 \beta q^{97} - 38556 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 306 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 306 q^{9} - 504 q^{11} - 440 q^{19} - 2376 q^{21} + 13860 q^{29} - 13504 q^{31} - 4752 q^{39} - 396 q^{41} - 26486 q^{49} + 27456 q^{51} - 49320 q^{59} - 11396 q^{61} - 96888 q^{69} - 106704 q^{71} + 103840 q^{79} - 145638 q^{81} + 19980 q^{89} + 14256 q^{91} - 77112 q^{99}+O(q^{100}) \) 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
−3.31662
3.31662
0 −19.8997 0 0 0 59.6992 0 153.000 0
1.2 0 19.8997 0 0 0 −59.6992 0 153.000 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 400.6.a.t 2
4.b odd 2 1 25.6.a.c 2
5.b even 2 1 inner 400.6.a.t 2
5.c odd 4 2 80.6.c.a 2
12.b even 2 1 225.6.a.n 2
15.e even 4 2 720.6.f.f 2
20.d odd 2 1 25.6.a.c 2
20.e even 4 2 5.6.b.a 2
40.i odd 4 2 320.6.c.g 2
40.k even 4 2 320.6.c.f 2
60.h even 2 1 225.6.a.n 2
60.l odd 4 2 45.6.b.b 2
140.j odd 4 2 245.6.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 20.e even 4 2
25.6.a.c 2 4.b odd 2 1
25.6.a.c 2 20.d odd 2 1
45.6.b.b 2 60.l odd 4 2
80.6.c.a 2 5.c odd 4 2
225.6.a.n 2 12.b even 2 1
225.6.a.n 2 60.h even 2 1
245.6.b.a 2 140.j odd 4 2
320.6.c.f 2 40.k even 4 2
320.6.c.g 2 40.i odd 4 2
400.6.a.t 2 1.a even 1 1 trivial
400.6.a.t 2 5.b even 2 1 inner
720.6.f.f 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 396 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 396 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3564 \) Copy content Toggle raw display
$11$ \( (T + 252)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 14256 \) Copy content Toggle raw display
$17$ \( T^{2} - 475904 \) Copy content Toggle raw display
$19$ \( (T + 220)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 5926316 \) Copy content Toggle raw display
$29$ \( (T - 6930)^{2} \) Copy content Toggle raw display
$31$ \( (T + 6752)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 195150384 \) Copy content Toggle raw display
$41$ \( (T + 198)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 174636 \) Copy content Toggle raw display
$47$ \( T^{2} - 111096524 \) Copy content Toggle raw display
$53$ \( T^{2} - 33918896 \) Copy content Toggle raw display
$59$ \( (T + 24660)^{2} \) Copy content Toggle raw display
$61$ \( (T + 5698)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 1904462604 \) Copy content Toggle raw display
$71$ \( (T + 53352)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 5030030016 \) Copy content Toggle raw display
$79$ \( (T - 51920)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 3824406476 \) Copy content Toggle raw display
$89$ \( (T - 9990)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 10251546624 \) Copy content Toggle raw display
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