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Properties

Label 400.2.y.c
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: 8.0.58140625.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{3} + 1) q^{3} + (\beta_{6} - \beta_{5} + \cdots + \beta_{3}) q^{5}+ \cdots + (\beta_{5} - \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{3} + 1) q^{3} + (\beta_{6} - \beta_{5} + \cdots + \beta_{3}) q^{5}+ \cdots + (2 \beta_{7} + 4 \beta_{5} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{3} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5 q^{3} + q^{9} + 4 q^{11} - 5 q^{13} - 15 q^{15} - 10 q^{17} + 5 q^{19} - 4 q^{21} - 5 q^{23} - 10 q^{25} + 5 q^{27} - 5 q^{29} + 9 q^{31} + 10 q^{33} - 15 q^{35} + 30 q^{37} + 3 q^{39} - 4 q^{41} - 15 q^{45} + 14 q^{49} + 4 q^{51} - 10 q^{53} + 10 q^{55} - 9 q^{61} - 10 q^{63} + 5 q^{65} - 20 q^{67} + 17 q^{69} - 6 q^{71} + 15 q^{73} + 10 q^{75} + 10 q^{77} - 15 q^{79} + 28 q^{81} + 45 q^{83} - 15 q^{85} + 20 q^{87} - 25 q^{89} - 6 q^{91} - 15 q^{95} - 60 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 420\nu^{7} - 776\nu^{6} + 698\nu^{5} - 1924\nu^{4} + 2297\nu^{3} + 5129\nu^{2} + 1055\nu - 10265 ) / 1355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 728\nu^{7} - 1327\nu^{6} + 1246\nu^{5} - 3353\nu^{4} + 3584\nu^{3} + 8547\nu^{2} + 2190\nu - 15715 ) / 1355 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -857\nu^{7} + 1666\nu^{6} - 1743\nu^{5} + 4424\nu^{4} - 4907\nu^{3} - 9470\nu^{2} - 2485\nu + 18200 ) / 1355 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 891\nu^{7} - 1623\nu^{6} + 1624\nu^{5} - 4492\nu^{4} + 4991\nu^{3} + 9520\nu^{2} + 3235\nu - 18960 ) / 1355 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 955\nu^{7} - 1829\nu^{6} + 1942\nu^{5} - 4891\nu^{4} + 5723\nu^{3} + 9646\nu^{2} + 2415\nu - 20550 ) / 1355 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{7} - 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{6} - 7\beta_{3} + 11\beta_{2} + 4\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 21\beta_{2} - 2\beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{7} + 38\beta_{5} + 23\beta_{4} - 23\beta_{3} + 12\beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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
1.17421 + 0.0566033i
−0.983224 − 0.644389i
1.66637 − 0.917186i
−0.357358 + 1.86824i
1.66637 + 0.917186i
−0.357358 − 1.86824i
1.17421 − 0.0566033i
−0.983224 + 0.644389i
0 −1.29224 + 1.77862i 0 −1.22570 + 1.87020i 0 − 0.992398i 0 −0.566541 − 1.74363i 0
129.2 0 0.865190 − 1.19083i 0 0.107666 − 2.23347i 0 − 3.26086i 0 0.257524 + 0.792578i 0
209.1 0 0.451659 − 0.146753i 0 2.19625 + 0.420099i 0 3.03582i 0 −2.24459 + 1.63079i 0
209.2 0 2.47539 − 0.804303i 0 −1.07822 − 1.95894i 0 − 0.407162i 0 3.05361 − 2.21858i 0
289.1 0 0.451659 + 0.146753i 0 2.19625 − 0.420099i 0 − 3.03582i 0 −2.24459 − 1.63079i 0
289.2 0 2.47539 + 0.804303i 0 −1.07822 + 1.95894i 0 0.407162i 0 3.05361 + 2.21858i 0
369.1 0 −1.29224 − 1.77862i 0 −1.22570 − 1.87020i 0 0.992398i 0 −0.566541 + 1.74363i 0
369.2 0 0.865190 + 1.19083i 0 0.107666 + 2.23347i 0 3.26086i 0 0.257524 − 0.792578i 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.c 8
4.b odd 2 1 25.2.e.a ✓ 8
12.b even 2 1 225.2.m.a 8
20.d odd 2 1 125.2.e.b 8
20.e even 4 2 125.2.d.b 16
25.e even 10 1 inner 400.2.y.c 8
25.f odd 20 2 10000.2.a.bj 8
100.h odd 10 1 25.2.e.a ✓ 8
100.h odd 10 1 625.2.b.c 8
100.h odd 10 1 625.2.e.a 8
100.h odd 10 1 625.2.e.i 8
100.j odd 10 1 125.2.e.b 8
100.j odd 10 1 625.2.b.c 8
100.j odd 10 1 625.2.e.a 8
100.j odd 10 1 625.2.e.i 8
100.l even 20 2 125.2.d.b 16
100.l even 20 2 625.2.a.f 8
100.l even 20 4 625.2.d.o 16
300.r even 10 1 225.2.m.a 8
300.u odd 20 2 5625.2.a.x 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a ✓ 8 4.b odd 2 1
25.2.e.a ✓ 8 100.h odd 10 1
125.2.d.b 16 20.e even 4 2
125.2.d.b 16 100.l even 20 2
125.2.e.b 8 20.d odd 2 1
125.2.e.b 8 100.j odd 10 1
225.2.m.a 8 12.b even 2 1
225.2.m.a 8 300.r even 10 1
400.2.y.c 8 1.a even 1 1 trivial
400.2.y.c 8 25.e even 10 1 inner
625.2.a.f 8 100.l even 20 2
625.2.b.c 8 100.h odd 10 1
625.2.b.c 8 100.j odd 10 1
625.2.d.o 16 100.l even 20 4
625.2.e.a 8 100.h odd 10 1
625.2.e.a 8 100.j odd 10 1
625.2.e.i 8 100.h odd 10 1
625.2.e.i 8 100.j odd 10 1
5625.2.a.x 8 300.u odd 20 2
10000.2.a.bj 8 25.f odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 5T_{3}^{7} + 9T_{3}^{6} - 15T_{3}^{5} + 51T_{3}^{4} - 110T_{3}^{3} + 144T_{3}^{2} - 80T_{3} + 16 \) 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
$3$ \( T^{8} - 5 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 21 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 5 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} + 10 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{8} + 5 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{8} + 5 T^{7} + \cdots + 483025 \) Copy content Toggle raw display
$31$ \( T^{8} - 9 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$37$ \( T^{8} - 30 T^{7} + \cdots + 116281 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + \cdots + 13456 \) Copy content Toggle raw display
$43$ \( T^{8} + 129 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + \cdots + 8755681 \) Copy content Toggle raw display
$59$ \( T^{8} - 15 T^{5} + \cdots + 4080400 \) Copy content Toggle raw display
$61$ \( T^{8} + 9 T^{7} + \cdots + 116281 \) Copy content Toggle raw display
$67$ \( T^{8} + 20 T^{7} + \cdots + 246016 \) Copy content Toggle raw display
$71$ \( T^{8} + 6 T^{7} + \cdots + 24245776 \) Copy content Toggle raw display
$73$ \( T^{8} - 15 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} + 15 T^{7} + \cdots + 33408400 \) Copy content Toggle raw display
$83$ \( T^{8} - 45 T^{7} + \cdots + 99856 \) Copy content Toggle raw display
$89$ \( T^{8} + 25 T^{7} + \cdots + 1392400 \) Copy content Toggle raw display
$97$ \( T^{8} + 60 T^{7} + \cdots + 301334881 \) Copy content Toggle raw display
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