Properties

Label 100.3.h.a
Level $100$
Weight $3$
Character orbit 100.h
Analytic conductor $2.725$
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] = [100,3,Mod(19,100)]
 
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, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 100.h (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: \(2.72480264360\)
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: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 4 \beta_{2} q^{4} + (3 \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 3) q^{5} + 4 \beta_{3} q^{8} - 9 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 4 \beta_{2} q^{4} + (3 \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 3) q^{5} + 4 \beta_{3} q^{8} - 9 \beta_{4} q^{9} + (3 \beta_{7} - 3 \beta_{5} + 8 \beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{10} + ( - 6 \beta_{7} + 5 \beta_{6} - 5 \beta_{2} + 6 \beta_1) q^{13} + 16 \beta_{4} q^{16} + (4 \beta_{5} - 15 \beta_{4} - 4 \beta_{3} - 15 \beta_{2} + 4 \beta_1) q^{17} - 9 \beta_{5} q^{18} + (8 \beta_{5} - 12) q^{20} + (7 \beta_{6} - 12 \beta_1) q^{25} + (5 \beta_{7} - 24 \beta_{6} + 24 \beta_{4} - 5 \beta_{3} + 24) q^{26} + ( - 21 \beta_{4} - 10 \beta_{3} + 21 \beta_{2} - 10 \beta_1 - 21) q^{29} + 16 \beta_{5} q^{32} + (16 \beta_{6} - 15 \beta_{5} - 16 \beta_{4} - 15 \beta_{3} + 16 \beta_{2}) q^{34} - 36 \beta_{6} q^{36} + ( - 35 \beta_{6} + 6 \beta_{5} + 35 \beta_{4} - 6 \beta_{3} - 35 \beta_{2} + \cdots + 70) q^{37}+ \cdots - 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 6 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 6 q^{5} + 18 q^{9} - 16 q^{10} - 32 q^{16} - 96 q^{20} + 14 q^{25} + 96 q^{26} - 84 q^{29} + 96 q^{34} - 72 q^{36} + 350 q^{37} + 64 q^{40} + 36 q^{41} + 54 q^{45} - 392 q^{49} - 96 q^{50} - 450 q^{53} - 44 q^{61} + 128 q^{64} + 438 q^{65} + 96 q^{74} - 96 q^{80} - 162 q^{81} + 258 q^{85} - 234 q^{89} + 144 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

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

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−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
−1.90211 + 0.618034i 0 3.23607 2.35114i −4.77819 + 1.47271i 0 0 −4.70228 + 6.47214i −2.78115 + 8.55951i 8.17848 5.75435i
19.2 1.90211 0.618034i 0 3.23607 2.35114i −0.0759100 4.99942i 0 0 4.70228 6.47214i −2.78115 + 8.55951i −3.23420 9.46255i
39.1 −1.17557 1.61803i 0 −1.23607 + 3.80423i 4.73128 + 1.61710i 0 0 7.60845 2.47214i 7.28115 + 5.29007i −2.94542 9.55638i
39.2 1.17557 + 1.61803i 0 −1.23607 + 3.80423i −2.87718 + 4.08924i 0 0 −7.60845 + 2.47214i 7.28115 + 5.29007i −9.99885 + 0.151820i
59.1 −1.17557 + 1.61803i 0 −1.23607 3.80423i 4.73128 1.61710i 0 0 7.60845 + 2.47214i 7.28115 5.29007i −2.94542 + 9.55638i
59.2 1.17557 1.61803i 0 −1.23607 3.80423i −2.87718 4.08924i 0 0 −7.60845 2.47214i 7.28115 5.29007i −9.99885 0.151820i
79.1 −1.90211 0.618034i 0 3.23607 + 2.35114i −4.77819 1.47271i 0 0 −4.70228 6.47214i −2.78115 8.55951i 8.17848 + 5.75435i
79.2 1.90211 + 0.618034i 0 3.23607 + 2.35114i −0.0759100 + 4.99942i 0 0 4.70228 + 6.47214i −2.78115 8.55951i −3.23420 + 9.46255i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
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.e even 10 1 inner
100.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.3.h.a 8
4.b odd 2 1 CM 100.3.h.a 8
25.e even 10 1 inner 100.3.h.a 8
100.h odd 10 1 inner 100.3.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.3.h.a 8 1.a even 1 1 trivial
100.3.h.a 8 4.b odd 2 1 CM
100.3.h.a 8 25.e even 10 1 inner
100.3.h.a 8 100.h odd 10 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{6} + 16 T^{4} - 64 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + 11 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 576 T^{6} + \cdots + 147355321 \) Copy content Toggle raw display
$17$ \( T^{8} - 256 T^{6} + \cdots + 12818994841 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 84 T^{7} + \cdots + 592159491361 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 350 T^{7} + \cdots + 33175998739321 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 132493014534721 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 450 T^{7} + \cdots + 27476268051961 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 44 T^{7} + \cdots + 36\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 347243062222681 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} + 234 T^{7} + \cdots + 12\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{8} - 20736 T^{6} + \cdots + 10\!\cdots\!61 \) Copy content Toggle raw display
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