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Properties

Label 100.5.b.e
Level $100$
Weight $5$
Character orbit 100.b
Analytic conductor $10.337$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,5,Mod(51,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.51");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 100.b (of order \(2\), degree \(1\), 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: \(10.3369963084\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1816805376000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 31x^{4} - 96x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{6} + 6) q^{4} + ( - \beta_{5} - \beta_{4}) q^{6} + (2 \beta_{3} + 6 \beta_{2} + \beta_1) q^{7} + ( - \beta_{7} + 3 \beta_{3} + \cdots + 2 \beta_1) q^{8} + ( - 3 \beta_{6} - 3 \beta_{4} - 39) q^{9}+ \cdots + (648 \beta_{6} - 252 \beta_{5} - 216 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 48 q^{4} - 312 q^{9} - 640 q^{14} - 832 q^{16} - 960 q^{21} - 1920 q^{24} + 2496 q^{26} - 2704 q^{29} + 2176 q^{34} - 5712 q^{36} + 1456 q^{41} - 1920 q^{44} + 7040 q^{46} + 8008 q^{49} + 11520 q^{54}+ \cdots - 76800 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 31x^{4} - 96x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 10\nu^{5} - \nu^{3} + 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 6\nu^{5} + 31\nu^{3} - 96\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 6\nu^{5} + 31\nu^{3} + 160\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} + \nu^{2} + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{4} - \nu^{2} + 28 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 31\nu^{2} + 72 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 10\nu^{5} - 13\nu^{3} + 72\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{4} + 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{3} + 8\beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + 2\beta_{4} - 26 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} - 7\beta_{3} + 11\beta_{2} + 20\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{6} + 12\beta_{5} - 7\beta_{4} - 108 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -19\beta_{7} - 8\beta_{3} - 22\beta_{2} + 58\beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.88164 − 0.677813i
1.88164 + 0.677813i
1.39980 − 1.42849i
1.39980 + 1.42849i
−1.39980 − 1.42849i
−1.39980 + 1.42849i
−1.88164 − 0.677813i
−1.88164 + 0.677813i
−3.76328 − 1.35563i − 13.9962i 12.3246 + 10.2032i 0 −18.9737 + 52.6718i − 35.6863i −32.5490 − 55.1050i −114.895 0
51.2 −3.76328 + 1.35563i 13.9962i 12.3246 − 10.2032i 0 −18.9737 − 52.6718i 35.6863i −32.5490 + 55.1050i −114.895 0
51.3 −2.79959 − 2.85697i 6.64118i −0.324555 + 15.9967i 0 18.9737 − 18.5926i − 39.0703i 46.6107 − 43.8570i 36.8947 0
51.4 −2.79959 + 2.85697i − 6.64118i −0.324555 − 15.9967i 0 18.9737 + 18.5926i 39.0703i 46.6107 + 43.8570i 36.8947 0
51.5 2.79959 − 2.85697i 6.64118i −0.324555 − 15.9967i 0 18.9737 + 18.5926i − 39.0703i −46.6107 − 43.8570i 36.8947 0
51.6 2.79959 + 2.85697i − 6.64118i −0.324555 + 15.9967i 0 18.9737 − 18.5926i 39.0703i −46.6107 + 43.8570i 36.8947 0
51.7 3.76328 − 1.35563i − 13.9962i 12.3246 − 10.2032i 0 −18.9737 − 52.6718i − 35.6863i 32.5490 − 55.1050i −114.895 0
51.8 3.76328 + 1.35563i 13.9962i 12.3246 + 10.2032i 0 −18.9737 + 52.6718i 35.6863i 32.5490 + 55.1050i −114.895 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.5.b.e 8
4.b odd 2 1 inner 100.5.b.e 8
5.b even 2 1 inner 100.5.b.e 8
5.c odd 4 2 20.5.d.c ✓ 8
15.e even 4 2 180.5.f.g 8
20.d odd 2 1 inner 100.5.b.e 8
20.e even 4 2 20.5.d.c ✓ 8
40.i odd 4 2 320.5.h.f 8
40.k even 4 2 320.5.h.f 8
60.l odd 4 2 180.5.f.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.d.c ✓ 8 5.c odd 4 2
20.5.d.c ✓ 8 20.e even 4 2
100.5.b.e 8 1.a even 1 1 trivial
100.5.b.e 8 4.b odd 2 1 inner
100.5.b.e 8 5.b even 2 1 inner
100.5.b.e 8 20.d odd 2 1 inner
180.5.f.g 8 15.e even 4 2
180.5.f.g 8 60.l odd 4 2
320.5.h.f 8 40.i odd 4 2
320.5.h.f 8 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{4} + 240T_{3}^{2} + 8640 \) Copy content Toggle raw display
\( T_{13}^{4} - 20928T_{13}^{2} + 82861056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 24 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{4} + 240 T^{2} + 8640)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2800 T^{2} + 1944000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 48000 T^{2} + 552407040)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 20928 T^{2} + 82861056)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 26368 T^{2} + 16367616)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 447360 T^{2} + 10372976640)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 350320 T^{2} + 30678152640)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 676 T + 104004)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1013760 T^{2} + 1745879040)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 3008448 T^{2} + 126031666176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 364 T - 3961116)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 10034800 T^{2} + 325194264000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 751600 T^{2} + 49724936640)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 20248378776576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 225559394979840)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 1644 T - 10307356)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 7632038946240)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 13443377725440)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 152966937968640)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 10\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6916 T - 62026236)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
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