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Properties

Label 100.3.d.b
Level $100$
Weight $3$
Character orbit 100.d
Analytic conductor $2.725$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [100,3,Mod(99,100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.72480264360\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content 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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{4} + 1) q^{4} + (2 \beta_{5} - \beta_{4} - \beta_{3}) q^{6} + (\beta_{7} + \beta_{6} + \cdots - 3 \beta_1) q^{7} + (\beta_{7} + 2 \beta_{6} + \cdots + \beta_1) q^{8}+ \cdots + (\beta_{5} + 10 \beta_{4} + 11 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 8 q^{9} - 80 q^{14} - 32 q^{16} + 80 q^{21} + 160 q^{24} - 24 q^{26} + 16 q^{29} - 184 q^{34} - 72 q^{36} - 224 q^{41} + 160 q^{44} + 80 q^{46} + 8 q^{49} - 160 q^{54} + 160 q^{56} + 256 q^{61}+ \cdots + 240 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

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

\(\zeta_{20}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( ( \beta_{4} + 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + \beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( ( \beta_{5} - 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( ( \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( ( \beta_{3} + 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{2} + \beta_1 ) / 8 \) 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 \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−0.951057 − 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 −2.35114 3.23607 + 2.35114i 0 4.47214 + 1.45309i 5.25731 −4.70228 − 6.47214i −3.47214 0
99.2 −1.90211 + 0.618034i −2.35114 3.23607 − 2.35114i 0 4.47214 − 1.45309i 5.25731 −4.70228 + 6.47214i −3.47214 0
99.3 −1.17557 − 1.61803i 3.80423 −1.23607 + 3.80423i 0 −4.47214 − 6.15537i 8.50651 7.60845 − 2.47214i 5.47214 0
99.4 −1.17557 + 1.61803i 3.80423 −1.23607 − 3.80423i 0 −4.47214 + 6.15537i 8.50651 7.60845 + 2.47214i 5.47214 0
99.5 1.17557 − 1.61803i −3.80423 −1.23607 − 3.80423i 0 −4.47214 + 6.15537i −8.50651 −7.60845 − 2.47214i 5.47214 0
99.6 1.17557 + 1.61803i −3.80423 −1.23607 + 3.80423i 0 −4.47214 − 6.15537i −8.50651 −7.60845 + 2.47214i 5.47214 0
99.7 1.90211 − 0.618034i 2.35114 3.23607 − 2.35114i 0 4.47214 − 1.45309i −5.25731 4.70228 − 6.47214i −3.47214 0
99.8 1.90211 + 0.618034i 2.35114 3.23607 + 2.35114i 0 4.47214 + 1.45309i −5.25731 4.70228 + 6.47214i −3.47214 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.b 8
3.b odd 2 1 900.3.f.e 8
4.b odd 2 1 inner 100.3.d.b 8
5.b even 2 1 inner 100.3.d.b 8
5.c odd 4 1 20.3.b.a ✓ 4
5.c odd 4 1 100.3.b.f 4
8.b even 2 1 1600.3.h.n 8
8.d odd 2 1 1600.3.h.n 8
12.b even 2 1 900.3.f.e 8
15.d odd 2 1 900.3.f.e 8
15.e even 4 1 180.3.c.a 4
15.e even 4 1 900.3.c.k 4
20.d odd 2 1 inner 100.3.d.b 8
20.e even 4 1 20.3.b.a ✓ 4
20.e even 4 1 100.3.b.f 4
40.e odd 2 1 1600.3.h.n 8
40.f even 2 1 1600.3.h.n 8
40.i odd 4 1 320.3.b.c 4
40.i odd 4 1 1600.3.b.s 4
40.k even 4 1 320.3.b.c 4
40.k even 4 1 1600.3.b.s 4
60.h even 2 1 900.3.f.e 8
60.l odd 4 1 180.3.c.a 4
60.l odd 4 1 900.3.c.k 4
80.i odd 4 1 1280.3.g.e 8
80.j even 4 1 1280.3.g.e 8
80.s even 4 1 1280.3.g.e 8
80.t odd 4 1 1280.3.g.e 8
120.q odd 4 1 2880.3.e.e 4
120.w even 4 1 2880.3.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a ✓ 4 5.c odd 4 1
20.3.b.a ✓ 4 20.e even 4 1
100.3.b.f 4 5.c odd 4 1
100.3.b.f 4 20.e even 4 1
100.3.d.b 8 1.a even 1 1 trivial
100.3.d.b 8 4.b odd 2 1 inner
100.3.d.b 8 5.b even 2 1 inner
100.3.d.b 8 20.d odd 2 1 inner
180.3.c.a 4 15.e even 4 1
180.3.c.a 4 60.l odd 4 1
320.3.b.c 4 40.i odd 4 1
320.3.b.c 4 40.k even 4 1
900.3.c.k 4 15.e even 4 1
900.3.c.k 4 60.l odd 4 1
900.3.f.e 8 3.b odd 2 1
900.3.f.e 8 12.b even 2 1
900.3.f.e 8 15.d odd 2 1
900.3.f.e 8 60.h even 2 1
1280.3.g.e 8 80.i odd 4 1
1280.3.g.e 8 80.j even 4 1
1280.3.g.e 8 80.s even 4 1
1280.3.g.e 8 80.t odd 4 1
1600.3.b.s 4 40.i odd 4 1
1600.3.b.s 4 40.k even 4 1
1600.3.h.n 8 8.b even 2 1
1600.3.h.n 8 8.d odd 2 1
1600.3.h.n 8 40.e odd 2 1
1600.3.h.n 8 40.f even 2 1
2880.3.e.e 4 120.q odd 4 1
2880.3.e.e 4 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 20T_{3}^{2} + 80 \) acting on \(S_{3}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} - 20 T^{2} + 80)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 100 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 400 T^{2} + 1280)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 72 T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 712 T^{2} + 80656)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 320 T^{2} + 20480)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 260 T^{2} + 80)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 76)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2320 T^{2} + 154880)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1032 T^{2} + 234256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 56 T + 604)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 500 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4100 T^{2} + 3561680)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 4872 T^{2} + 2062096)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 5760 T^{2} + 1658880)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 64 T - 2356)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 10420 T^{2} + 19920080)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8080 T^{2} + 10138880)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 18952 T^{2} + 583696)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 13120 T^{2} + 2478080)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 6260 T^{2} + 2620880)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 44 T - 7516)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 10152 T^{2} + 13220496)^{2} \) Copy content Toggle raw display
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