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Properties

Label 100.6.c.b
Level $100$
Weight $6$
Character orbit 100.c
Analytic conductor $16.038$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [100,6,Mod(49,100)] 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("100.49"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 100.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,198] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.0383819813\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 6 \beta q^{3} - 44 \beta q^{7} + 99 q^{9} + 540 q^{11} + 209 \beta q^{13} + 297 \beta q^{17} - 836 q^{19} + 1056 q^{21} + 2052 \beta q^{23} + 2052 \beta q^{27} + 594 q^{29} + 4256 q^{31} + 3240 \beta q^{33} + \cdots + 53460 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 198 q^{9} + 1080 q^{11} - 1672 q^{19} + 2112 q^{21} + 1188 q^{29} + 8512 q^{31} - 10032 q^{39} + 34452 q^{41} + 18126 q^{49} - 14256 q^{51} + 15336 q^{59} - 69476 q^{61} - 98496 q^{69} - 93744 q^{71}+ \cdots + 106920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 1.00000i
1.00000i
0 − 12.0000i 0 0 0 88.0000i 0 99.0000 0
49.2 0 12.0000i 0 0 0 − 88.0000i 0 99.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.6.c.b 2
3.b odd 2 1 900.6.d.a 2
4.b odd 2 1 400.6.c.f 2
5.b even 2 1 inner 100.6.c.b 2
5.c odd 4 1 4.6.a.a ✓ 1
5.c odd 4 1 100.6.a.b 1
15.d odd 2 1 900.6.d.a 2
15.e even 4 1 36.6.a.a 1
15.e even 4 1 900.6.a.h 1
20.d odd 2 1 400.6.c.f 2
20.e even 4 1 16.6.a.b 1
20.e even 4 1 400.6.a.d 1
35.f even 4 1 196.6.a.e 1
35.k even 12 2 196.6.e.d 2
35.l odd 12 2 196.6.e.g 2
40.i odd 4 1 64.6.a.f 1
40.k even 4 1 64.6.a.b 1
45.k odd 12 2 324.6.e.a 2
45.l even 12 2 324.6.e.d 2
55.e even 4 1 484.6.a.a 1
60.l odd 4 1 144.6.a.c 1
65.f even 4 1 676.6.d.a 2
65.h odd 4 1 676.6.a.a 1
65.k even 4 1 676.6.d.a 2
80.i odd 4 1 256.6.b.g 2
80.j even 4 1 256.6.b.c 2
80.s even 4 1 256.6.b.c 2
80.t odd 4 1 256.6.b.g 2
120.q odd 4 1 576.6.a.bd 1
120.w even 4 1 576.6.a.bc 1
140.j odd 4 1 784.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a ✓ 1 5.c odd 4 1
16.6.a.b 1 20.e even 4 1
36.6.a.a 1 15.e even 4 1
64.6.a.b 1 40.k even 4 1
64.6.a.f 1 40.i odd 4 1
100.6.a.b 1 5.c odd 4 1
100.6.c.b 2 1.a even 1 1 trivial
100.6.c.b 2 5.b even 2 1 inner
144.6.a.c 1 60.l odd 4 1
196.6.a.e 1 35.f even 4 1
196.6.e.d 2 35.k even 12 2
196.6.e.g 2 35.l odd 12 2
256.6.b.c 2 80.j even 4 1
256.6.b.c 2 80.s even 4 1
256.6.b.g 2 80.i odd 4 1
256.6.b.g 2 80.t odd 4 1
324.6.e.a 2 45.k odd 12 2
324.6.e.d 2 45.l even 12 2
400.6.a.d 1 20.e even 4 1
400.6.c.f 2 4.b odd 2 1
400.6.c.f 2 20.d odd 2 1
484.6.a.a 1 55.e even 4 1
576.6.a.bc 1 120.w even 4 1
576.6.a.bd 1 120.q odd 4 1
676.6.a.a 1 65.h odd 4 1
676.6.d.a 2 65.f even 4 1
676.6.d.a 2 65.k even 4 1
784.6.a.d 1 140.j odd 4 1
900.6.a.h 1 15.e even 4 1
900.6.d.a 2 3.b odd 2 1
900.6.d.a 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 144 \) acting on \(S_{6}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 144 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 7744 \) Copy content Toggle raw display
$11$ \( (T - 540)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 174724 \) Copy content Toggle raw display
$17$ \( T^{2} + 352836 \) Copy content Toggle raw display
$19$ \( (T + 836)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16842816 \) Copy content Toggle raw display
$29$ \( (T - 594)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4256)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 88804 \) Copy content Toggle raw display
$41$ \( (T - 17226)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 146410000 \) Copy content Toggle raw display
$47$ \( T^{2} + 1679616 \) Copy content Toggle raw display
$53$ \( T^{2} + 380016036 \) Copy content Toggle raw display
$59$ \( (T - 7668)^{2} \) Copy content Toggle raw display
$61$ \( (T + 34738)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 475763344 \) Copy content Toggle raw display
$71$ \( (T + 46872)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4564623844 \) Copy content Toggle raw display
$79$ \( (T - 76912)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4585456656 \) Copy content Toggle raw display
$89$ \( (T + 29754)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 14981270404 \) Copy content Toggle raw display
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