Properties

Label 100.9.j.a
Level $100$
Weight $9$
Character orbit 100.j
Analytic conductor $40.738$
Analytic rank $0$
Dimension $8$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,9,Mod(11,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 8]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.11");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 100.j (of order \(10\), degree \(4\), 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: \(40.7378610061\)
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: \( 2^{16}\cdot 3^{4}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (16 \beta_{6} - 16 \beta_{4} + \cdots - 16) q^{2}+ \cdots - 6561 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (16 \beta_{6} - 16 \beta_{4} + \cdots - 16) q^{2}+ \cdots + (92236816 \beta_{6} - 92236816 \beta_{4} + \cdots - 92236816) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{2} - 512 q^{4} + 1054 q^{5} - 8192 q^{8} - 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{2} - 512 q^{4} + 1054 q^{5} - 8192 q^{8} - 13122 q^{9} + 16864 q^{10} + 956 q^{13} - 131072 q^{16} + 126716 q^{17} + 839808 q^{18} - 1079296 q^{20} - 329666 q^{25} + 15296 q^{26} + 2815676 q^{29} + 8388608 q^{32} - 3041184 q^{34} - 3359232 q^{36} + 2777766 q^{37} + 4317184 q^{40} - 7155844 q^{41} + 6915294 q^{45} + 46118408 q^{49} - 5274656 q^{50} + 244736 q^{52} - 28861914 q^{53} + 45050816 q^{58} - 41444164 q^{61} - 33554432 q^{64} + 96717318 q^{65} + 32439296 q^{68} - 53747712 q^{72} + 109434236 q^{73} - 29629504 q^{74} + 69074944 q^{80} - 86093442 q^{81} - 114493504 q^{82} + 359829798 q^{85} - 90797754 q^{89} + 110644704 q^{90} + 346759676 q^{97} - 184473632 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 336\zeta_{20} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{20}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 336\zeta_{20}^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 336\zeta_{20}^{5} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{20}^{6} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 336\zeta_{20}^{7} \) Copy content Toggle raw display
\(\zeta_{20}\)\(=\) \( ( \beta_1 ) / 336 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( \beta_{3} ) / 336 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( ( \beta_{5} ) / 336 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( \beta_{6} \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( \beta_{7} ) / 336 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.951057 0.309017i
0.951057 + 0.309017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.951057 + 0.309017i
0.951057 0.309017i
−12.9443 + 9.40456i 0 79.1084 243.470i −482.407 397.377i 0 0 1265.73 + 3895.53i −5307.96 3856.46i 9981.56 + 606.930i
11.2 −12.9443 + 9.40456i 0 79.1084 243.470i 156.703 605.036i 0 0 1265.73 + 3895.53i −5307.96 3856.46i 3661.70 + 9305.48i
31.1 4.94427 + 15.2169i 0 −207.108 + 150.473i 228.856 + 581.593i 0 0 −3313.73 2407.57i 2027.46 6239.88i −7718.51 + 6358.03i
31.2 4.94427 + 15.2169i 0 −207.108 + 150.473i 623.848 + 37.9331i 0 0 −3313.73 2407.57i 2027.46 6239.88i 2507.25 + 9680.58i
71.1 4.94427 15.2169i 0 −207.108 150.473i 228.856 581.593i 0 0 −3313.73 + 2407.57i 2027.46 + 6239.88i −7718.51 6358.03i
71.2 4.94427 15.2169i 0 −207.108 150.473i 623.848 37.9331i 0 0 −3313.73 + 2407.57i 2027.46 + 6239.88i 2507.25 9680.58i
91.1 −12.9443 9.40456i 0 79.1084 + 243.470i −482.407 + 397.377i 0 0 1265.73 3895.53i −5307.96 + 3856.46i 9981.56 606.930i
91.2 −12.9443 9.40456i 0 79.1084 + 243.470i 156.703 + 605.036i 0 0 1265.73 3895.53i −5307.96 + 3856.46i 3661.70 9305.48i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
25.d even 5 1 inner
100.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.j.a 8
4.b odd 2 1 CM 100.9.j.a 8
25.d even 5 1 inner 100.9.j.a 8
100.j odd 10 1 inner 100.9.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.9.j.a 8 1.a even 1 1 trivial
100.9.j.a 8 4.b odd 2 1 CM
100.9.j.a 8 25.d even 5 1 inner
100.9.j.a 8 100.j odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{9}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 16 T^{3} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 49\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 22\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 80\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 74\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 18\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 20\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 36\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
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