Properties

Label 400.2.y.a
Level $400$
Weight $2$
Character orbit 400.y
Analytic conductor $3.194$
Analytic rank $0$
Dimension $8$
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(129,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(10))
 
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("400.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.y (of order \(10\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}^{2} + \zeta_{20}) q^{3} + (\zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} + \zeta_{20} - 2) q^{5} + (2 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 1) q^{7} + (2 \zeta_{20}^{5} + \zeta_{20}^{4} - \zeta_{20}^{2} - 2 \zeta_{20}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}^{2} + \zeta_{20}) q^{3} + (\zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} + \zeta_{20} - 2) q^{5} + (2 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 1) q^{7} + (2 \zeta_{20}^{5} + \zeta_{20}^{4} - \zeta_{20}^{2} - 2 \zeta_{20}) q^{9} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} + \zeta_{20}^{4} - \zeta_{20}^{3} - \zeta_{20}^{2} - 3 \zeta_{20} + 1) q^{11} + (\zeta_{20}^{5} - \zeta_{20}^{4} - 4 \zeta_{20}^{3} - \zeta_{20}^{2} + \zeta_{20}) q^{13} + (2 \zeta_{20}^{7} - \zeta_{20}^{4} - 2 \zeta_{20}^{3} + \zeta_{20}^{2} + 2) q^{15} + (\zeta_{20}^{2} - 2 \zeta_{20} + 1) q^{17} + ( - \zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{5} + 2 \zeta_{20}^{2} - 2) q^{19} + ( - \zeta_{20}^{7} + 3 \zeta_{20}^{6} - \zeta_{20}^{5} - 2 \zeta_{20}^{4} - \zeta_{20}^{3} + 3 \zeta_{20}^{2} - \zeta_{20}) q^{21} + (2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{3} + \zeta_{20}^{2} - 2 \zeta_{20} - 2) q^{23} + ( - \zeta_{20}^{6} - 4 \zeta_{20}^{5} - \zeta_{20}^{4} + 2 \zeta_{20}^{3} + \zeta_{20}^{2} - 4 \zeta_{20} + 1) q^{25} + ( - \zeta_{20}^{7} - 2 \zeta_{20}^{6} + \zeta_{20}^{4} + 3 \zeta_{20}^{3} - \zeta_{20}^{2} - 3 \zeta_{20} + 1) q^{27} + ( - \zeta_{20}^{7} + 3 \zeta_{20}^{6} + \zeta_{20}^{4} + 3 \zeta_{20}^{2} - \zeta_{20}) q^{29} + (2 \zeta_{20}^{7} + 3 \zeta_{20}^{6} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{2} - 2) q^{31} + ( - 4 \zeta_{20}^{7} + 4 \zeta_{20}^{5} - 3 \zeta_{20}^{2} + 4 \zeta_{20} - 3) q^{33} + (\zeta_{20}^{7} + \zeta_{20}^{6} - 6 \zeta_{20}^{5} + 3 \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 3 \zeta_{20}^{2} - \zeta_{20}) q^{35} + ( - \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} + 5 \zeta_{20}^{3} + \zeta_{20}^{2} - \zeta_{20} - 1) q^{37} + ( - 4 \zeta_{20}^{7} + 8 \zeta_{20}^{5} + 3 \zeta_{20}^{4} - 6 \zeta_{20}^{3} - 5 \zeta_{20}^{2} + 2 \zeta_{20} + 3) q^{39} + (8 \zeta_{20}^{7} - \zeta_{20}^{6} - 2 \zeta_{20}^{5} + 5 \zeta_{20}^{4} + 4 \zeta_{20}^{3} - 5 \zeta_{20}^{2} - 6 \zeta_{20} + 1) q^{41} + ( - 3 \zeta_{20}^{7} - \zeta_{20}^{6} - 2 \zeta_{20}^{5} - 5 \zeta_{20}^{4} - 3 \zeta_{20}^{3} + 4 \zeta_{20}^{2} + \cdots - 2) q^{43}+ \cdots + ( - 2 \zeta_{20}^{7} - 7 \zeta_{20}^{6} - \zeta_{20}^{5} + 7 \zeta_{20}^{4} + 4 \zeta_{20}^{3} - 2 \zeta_{20} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{5} - 4 q^{9} + 4 q^{11} + 20 q^{15} + 10 q^{17} - 10 q^{19} + 16 q^{21} - 10 q^{23} + 10 q^{25} + 10 q^{29} - 6 q^{31} - 30 q^{33} - 10 q^{35} - 10 q^{37} + 8 q^{39} - 14 q^{41} + 10 q^{45} + 30 q^{47} - 16 q^{49} - 16 q^{51} + 10 q^{55} - 14 q^{61} - 20 q^{63} + 50 q^{65} - 10 q^{67} - 8 q^{69} + 34 q^{71} - 10 q^{75} + 40 q^{77} - 12 q^{81} - 50 q^{83} - 20 q^{85} - 20 q^{87} + 20 q^{89} + 4 q^{91} - 20 q^{97} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-\zeta_{20}^{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
0 0.530249 0.729825i 0 −2.22982 0.166977i 0 5.07768i 0 0.675571 + 2.07919i 0
129.2 0 1.70582 2.34786i 0 0.847859 + 2.06909i 0 1.07768i 0 −1.67557 5.15688i 0
209.1 0 −2.06909 + 0.672288i 0 −2.17229 0.530249i 0 2.72654i 0 1.40211 1.01869i 0
209.2 0 −0.166977 + 0.0542543i 0 −1.44575 + 1.70582i 0 1.27346i 0 −2.40211 + 1.74524i 0
289.1 0 −2.06909 0.672288i 0 −2.17229 + 0.530249i 0 2.72654i 0 1.40211 + 1.01869i 0
289.2 0 −0.166977 0.0542543i 0 −1.44575 1.70582i 0 1.27346i 0 −2.40211 1.74524i 0
369.1 0 0.530249 + 0.729825i 0 −2.22982 + 0.166977i 0 5.07768i 0 0.675571 2.07919i 0
369.2 0 1.70582 + 2.34786i 0 0.847859 2.06909i 0 1.07768i 0 −1.67557 + 5.15688i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.y.a 8
4.b odd 2 1 50.2.e.a 8
12.b even 2 1 450.2.l.b 8
20.d odd 2 1 250.2.e.a 8
20.e even 4 1 250.2.d.b 8
20.e even 4 1 250.2.d.c 8
25.e even 10 1 inner 400.2.y.a 8
25.f odd 20 1 10000.2.a.o 4
25.f odd 20 1 10000.2.a.bb 4
100.h odd 10 1 50.2.e.a 8
100.h odd 10 1 1250.2.b.c 8
100.j odd 10 1 250.2.e.a 8
100.j odd 10 1 1250.2.b.c 8
100.l even 20 1 250.2.d.b 8
100.l even 20 1 250.2.d.c 8
100.l even 20 1 1250.2.a.h 4
100.l even 20 1 1250.2.a.i 4
300.r even 10 1 450.2.l.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.e.a 8 4.b odd 2 1
50.2.e.a 8 100.h odd 10 1
250.2.d.b 8 20.e even 4 1
250.2.d.b 8 100.l even 20 1
250.2.d.c 8 20.e even 4 1
250.2.d.c 8 100.l even 20 1
250.2.e.a 8 20.d odd 2 1
250.2.e.a 8 100.j odd 10 1
400.2.y.a 8 1.a even 1 1 trivial
400.2.y.a 8 25.e even 10 1 inner
450.2.l.b 8 12.b even 2 1
450.2.l.b 8 300.r even 10 1
1250.2.a.h 4 100.l even 20 1
1250.2.a.i 4 100.l even 20 1
1250.2.b.c 8 100.h odd 10 1
1250.2.b.c 8 100.j odd 10 1
10000.2.a.o 4 25.f odd 20 1
10000.2.a.bb 4 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - T_{3}^{6} + 20T_{3}^{5} + 26T_{3}^{4} - 20T_{3}^{3} + 24T_{3}^{2} + 10T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + 20 T^{5} + 26 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + 10 T^{7} + 45 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 36 T^{6} + 286 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} - 3 T^{6} + 58 T^{5} + \cdots + 3481 \) Copy content Toggle raw display
$13$ \( T^{8} - 31 T^{6} - 110 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$17$ \( T^{8} - 10 T^{7} + 41 T^{6} - 80 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} + 10 T^{7} + 60 T^{6} + 220 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{8} + 10 T^{7} + 59 T^{6} + 100 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{8} - 10 T^{7} + 105 T^{6} + \cdots + 9025 \) Copy content Toggle raw display
$31$ \( T^{8} + 6 T^{7} + 67 T^{6} + 48 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{8} + 10 T^{7} - 9 T^{6} + \cdots + 58081 \) Copy content Toggle raw display
$41$ \( T^{8} + 14 T^{7} + 127 T^{6} + \cdots + 28504921 \) Copy content Toggle raw display
$43$ \( T^{8} + 224 T^{6} + 16686 T^{4} + \cdots + 3682561 \) Copy content Toggle raw display
$47$ \( T^{8} - 30 T^{7} + 331 T^{6} + \cdots + 8637721 \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{6} - 160 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( T^{8} + 2560 T^{4} + \cdots + 1638400 \) Copy content Toggle raw display
$61$ \( T^{8} + 14 T^{7} + 152 T^{6} + \cdots + 32761 \) Copy content Toggle raw display
$67$ \( T^{8} + 10 T^{7} + 61 T^{6} + \cdots + 3481 \) Copy content Toggle raw display
$71$ \( T^{8} - 34 T^{7} + 567 T^{6} + \cdots + 1042441 \) Copy content Toggle raw display
$73$ \( T^{8} - 36 T^{6} - 2080 T^{5} + \cdots + 92416 \) Copy content Toggle raw display
$79$ \( T^{8} + 20 T^{6} + 80 T^{5} + \cdots + 22278400 \) Copy content Toggle raw display
$83$ \( T^{8} + 50 T^{7} + 1169 T^{6} + \cdots + 17131321 \) Copy content Toggle raw display
$89$ \( (T^{4} - 10 T^{3} + 160 T^{2} - 400 T + 400)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 20 T^{7} + 76 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
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