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Properties

Label 100.7.h.a
Level $100$
Weight $7$
Character orbit 100.h
Analytic conductor $23.005$
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] = [100,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(23.0054083620\)
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: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} + 64 \beta_{2} q^{4} + ( - 117 \beta_{6} + 117 \beta_{4} + \cdots + 117) q^{5}+ \cdots - 729 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{2} + 64 \beta_{2} q^{4} + ( - 117 \beta_{6} + 117 \beta_{4} + \cdots + 117) q^{5}+ \cdots - 235298 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{4} + 234 q^{5} + 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 128 q^{4} + 234 q^{5} + 1458 q^{9} + 704 q^{10} - 8192 q^{16} + 59904 q^{20} - 23506 q^{25} + 26496 q^{26} + 63756 q^{29} - 234624 q^{34} - 93312 q^{36} + 277550 q^{37} - 45056 q^{40} - 169884 q^{41} - 170586 q^{45} - 941192 q^{49} - 164736 q^{50} + 147150 q^{53} + 469876 q^{61} + 524288 q^{64} + 2162358 q^{65} - 1355904 q^{74} + 958464 q^{80} - 1062882 q^{81} - 1869582 q^{85} + 4136886 q^{89} - 513216 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{20}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{20}^j\)

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
−7.60845 + 2.47214i 0 51.7771 − 37.6183i 120.518 + 33.1741i 0 0 −300.946 + 414.217i −225.273 + 693.320i −998.963 + 45.5320i
19.2 7.60845 − 2.47214i 0 51.7771 − 37.6183i 68.7924 + 104.368i 0 0 300.946 − 414.217i −225.273 + 693.320i 781.415 + 624.012i
39.1 −4.70228 − 6.47214i 0 −19.7771 + 60.8676i −78.0015 − 97.6769i 0 0 486.941 − 158.217i 589.773 + 428.495i −265.393 + 964.140i
39.2 4.70228 + 6.47214i 0 −19.7771 + 60.8676i 5.69150 − 124.870i 0 0 −486.941 + 158.217i 589.773 + 428.495i 834.941 − 550.339i
59.1 −4.70228 + 6.47214i 0 −19.7771 − 60.8676i −78.0015 + 97.6769i 0 0 486.941 + 158.217i 589.773 − 428.495i −265.393 − 964.140i
59.2 4.70228 − 6.47214i 0 −19.7771 − 60.8676i 5.69150 + 124.870i 0 0 −486.941 − 158.217i 589.773 − 428.495i 834.941 + 550.339i
79.1 −7.60845 − 2.47214i 0 51.7771 + 37.6183i 120.518 − 33.1741i 0 0 −300.946 − 414.217i −225.273 − 693.320i −998.963 − 45.5320i
79.2 7.60845 + 2.47214i 0 51.7771 + 37.6183i 68.7924 − 104.368i 0 0 300.946 + 414.217i −225.273 − 693.320i 781.415 − 624.012i
\(n\): 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 100.7.h.a ✓ 8
4.b odd 2 1 CM 100.7.h.a ✓ 8
25.e even 10 1 inner 100.7.h.a ✓ 8
100.h odd 10 1 inner 100.7.h.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.7.h.a ✓ 8 1.a even 1 1 trivial
100.7.h.a ✓ 8 4.b odd 2 1 CM
100.7.h.a ✓ 8 25.e even 10 1 inner
100.7.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 \(S_{7}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 64 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 33\!\cdots\!61 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 48\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 58\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 42\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 44\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 71\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 15\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 53\!\cdots\!81 \) Copy content Toggle raw display
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