↑

Properties

Label 100.3.d.a
Level $100$
Weight $3$
Character orbit 100.d
Analytic conductor $2.725$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,3,Mod(99,100)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 100.d (of order \(2\), degree \(1\), 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: \(2.72480264360\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18084870400.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 40x^{4} + 17x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} + (\beta_{4} - 1) q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{6} + ( - 2 \beta_{7} + 2 \beta_{2}) q^{7} + (\beta_{7} + \beta_{6} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - 2 \beta_{4} + 2 \beta_{3} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} + (\beta_{4} - 1) q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{6} + ( - 2 \beta_{7} + 2 \beta_{2}) q^{7} + (\beta_{7} + \beta_{6} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - 6 \beta_{5} - 12 \beta_{4} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4} + 18 q^{6} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{4} + 18 q^{6} + 32 q^{9} + 52 q^{14} - 62 q^{16} - 64 q^{21} - 158 q^{24} + 168 q^{26} - 80 q^{29} + 158 q^{34} - 204 q^{36} + 136 q^{41} - 170 q^{44} + 68 q^{46} + 152 q^{49} + 74 q^{54} - 92 q^{56} - 224 q^{61} + 110 q^{64} + 10 q^{66} - 336 q^{69} + 228 q^{74} + 90 q^{76} + 504 q^{81} + 244 q^{84} + 128 q^{86} + 440 q^{89} - 408 q^{94} + 558 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} + 40x^{4} + 17x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 68\nu^{3} + 53\nu ) / 22 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{7} - 87\nu^{5} + 333\nu^{3} - 118\nu ) / 55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{6} - 227\nu^{4} + 903\nu^{2} - 98 ) / 55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\nu^{6} - 261\nu^{4} + 889\nu^{2} + 251 ) / 55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 39\nu^{6} - 469\nu^{4} + 1651\nu^{2} + 254 ) / 55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{7} + 104\nu^{5} - 326\nu^{3} - 249\nu ) / 22 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -79\nu^{7} + 974\nu^{5} - 3436\nu^{3} - 609\nu ) / 110 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - 5\beta_{2} - 5\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} - 8\beta_{4} + 3\beta_{3} + 28 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - 8\beta_{6} - 10\beta_{2} - 35\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{5} - 13\beta_{4} + 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 65\beta_{7} - 81\beta_{6} + 35\beta_{2} - 200\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 308\beta_{5} - 443\beta_{4} - 127\beta_{3} + 373 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 570\beta_{7} - 643\beta_{6} + 905\beta_{2} - 905\beta_1 ) / 10 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.220086 + 0.500000i
0.220086 − 0.500000i
2.53999 − 0.500000i
2.53999 + 0.500000i
−2.53999 − 0.500000i
−2.53999 + 0.500000i
−0.220086 + 0.500000i
−0.220086 − 0.500000i
−1.47492 − 1.35078i −0.440172 0.350781 + 3.98459i 0 0.649219 + 0.594576i −10.9190 4.86493 − 6.35078i −8.80625 0
99.2 −1.47492 + 1.35078i −0.440172 0.350781 − 3.98459i 0 0.649219 − 0.594576i −10.9190 4.86493 + 6.35078i −8.80625 0
99.3 −0.758030 − 1.85078i −5.07999 −2.85078 + 2.80590i 0 3.85078 + 9.40194i 4.09573 7.35408 + 3.14922i 16.8062 0
99.4 −0.758030 + 1.85078i −5.07999 −2.85078 − 2.80590i 0 3.85078 − 9.40194i 4.09573 7.35408 − 3.14922i 16.8062 0
99.5 0.758030 − 1.85078i 5.07999 −2.85078 − 2.80590i 0 3.85078 − 9.40194i −4.09573 −7.35408 + 3.14922i 16.8062 0
99.6 0.758030 + 1.85078i 5.07999 −2.85078 + 2.80590i 0 3.85078 + 9.40194i −4.09573 −7.35408 − 3.14922i 16.8062 0
99.7 1.47492 − 1.35078i 0.440172 0.350781 − 3.98459i 0 0.649219 − 0.594576i 10.9190 −4.86493 − 6.35078i −8.80625 0
99.8 1.47492 + 1.35078i 0.440172 0.350781 + 3.98459i 0 0.649219 + 0.594576i 10.9190 −4.86493 + 6.35078i −8.80625 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
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 100.3.d.a 8
3.b odd 2 1 900.3.f.d 8
4.b odd 2 1 inner 100.3.d.a 8
5.b even 2 1 inner 100.3.d.a 8
5.c odd 4 1 100.3.b.d ✓ 4
5.c odd 4 1 100.3.b.e yes 4
8.b even 2 1 1600.3.h.o 8
8.d odd 2 1 1600.3.h.o 8
12.b even 2 1 900.3.f.d 8
15.d odd 2 1 900.3.f.d 8
15.e even 4 1 900.3.c.l 4
15.e even 4 1 900.3.c.m 4
20.d odd 2 1 inner 100.3.d.a 8
20.e even 4 1 100.3.b.d ✓ 4
20.e even 4 1 100.3.b.e yes 4
40.e odd 2 1 1600.3.h.o 8
40.f even 2 1 1600.3.h.o 8
40.i odd 4 1 1600.3.b.o 4
40.i odd 4 1 1600.3.b.p 4
40.k even 4 1 1600.3.b.o 4
40.k even 4 1 1600.3.b.p 4
60.h even 2 1 900.3.f.d 8
60.l odd 4 1 900.3.c.l 4
60.l odd 4 1 900.3.c.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.3.b.d ✓ 4 5.c odd 4 1
100.3.b.d ✓ 4 20.e even 4 1
100.3.b.e yes 4 5.c odd 4 1
100.3.b.e yes 4 20.e even 4 1
100.3.d.a 8 1.a even 1 1 trivial
100.3.d.a 8 4.b odd 2 1 inner
100.3.d.a 8 5.b even 2 1 inner
100.3.d.a 8 20.d odd 2 1 inner
900.3.c.l 4 15.e even 4 1
900.3.c.l 4 60.l odd 4 1
900.3.c.m 4 15.e even 4 1
900.3.c.m 4 60.l odd 4 1
900.3.f.d 8 3.b odd 2 1
900.3.f.d 8 12.b even 2 1
900.3.f.d 8 15.d odd 2 1
900.3.f.d 8 60.h even 2 1
1600.3.b.o 4 40.i odd 4 1
1600.3.b.o 4 40.k even 4 1
1600.3.b.p 4 40.i odd 4 1
1600.3.b.p 4 40.k even 4 1
1600.3.h.o 8 8.b even 2 1
1600.3.h.o 8 8.d odd 2 1
1600.3.h.o 8 40.e odd 2 1
1600.3.h.o 8 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 26T_{3}^{2} + 5 \) acting on \(S_{3}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} - 26 T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 136 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 130 T^{2} + 125)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 336 T^{2} + 25600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 346 T^{2} + 24025)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1370 T^{2} + 465125)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 776 T^{2} + 147920)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 20 T - 64)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 520 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2376 T^{2} + 739600)^{2} \) Copy content Toggle raw display
$41$ \( (T - 17)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 3056 T^{2} + 128000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4976 T^{2} + 5242880)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8136 T^{2} + 1392400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 13680 T^{2} + 41472000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 56 T - 3316)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 3586 T^{2} + 1915805)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 19120 T^{2} + 67712000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 6826 T^{2} + 9517225)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 7720 T^{2} + 6962000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 5426 T^{2} + 3621005)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 110 T + 2861)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 23976 T^{2} + 32400)^{2} \) Copy content Toggle raw display
show more
show less