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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,2,Mod(49,450)] 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("450.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,6,0,0,-6,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{2} + 2) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} - 3 \zeta_{12}^{2} q^{9} + ( - 3 \zeta_{12}^{2} + 3) q^{11} + \cdots - 9 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 6 q^{6} - 6 q^{9} + 6 q^{11} + 4 q^{14} - 2 q^{16} + 4 q^{19} + 12 q^{21} + 6 q^{24} + 8 q^{26} + 12 q^{29} + 8 q^{31} - 6 q^{34} + 6 q^{36} - 18 q^{41} + 12 q^{44} - 24 q^{46} - 6 q^{49}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i −0.866025 − 1.50000i 0.500000 − 0.866025i 0 1.50000 + 0.866025i −1.73205 + 1.00000i 1.00000i −1.50000 + 2.59808i 0
49.2 0.866025 − 0.500000i 0.866025 + 1.50000i 0.500000 − 0.866025i 0 1.50000 + 0.866025i 1.73205 − 1.00000i − 1.00000i −1.50000 + 2.59808i 0
349.1 −0.866025 − 0.500000i −0.866025 + 1.50000i 0.500000 + 0.866025i 0 1.50000 − 0.866025i −1.73205 − 1.00000i − 1.00000i −1.50000 − 2.59808i 0
349.2 0.866025 + 0.500000i 0.866025 − 1.50000i 0.500000 + 0.866025i 0 1.50000 − 0.866025i 1.73205 + 1.00000i 1.00000i −1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.j.e 4
3.b odd 2 1 1350.2.j.a 4
5.b even 2 1 inner 450.2.j.e 4
5.c odd 4 1 18.2.c.a ✓ 2
5.c odd 4 1 450.2.e.i 2
9.c even 3 1 inner 450.2.j.e 4
9.c even 3 1 4050.2.c.c 2
9.d odd 6 1 1350.2.j.a 4
9.d odd 6 1 4050.2.c.r 2
15.d odd 2 1 1350.2.j.a 4
15.e even 4 1 54.2.c.a 2
15.e even 4 1 1350.2.e.c 2
20.e even 4 1 144.2.i.c 2
35.f even 4 1 882.2.f.d 2
35.k even 12 1 882.2.e.g 2
35.k even 12 1 882.2.h.b 2
35.l odd 12 1 882.2.e.i 2
35.l odd 12 1 882.2.h.c 2
40.i odd 4 1 576.2.i.g 2
40.k even 4 1 576.2.i.a 2
45.h odd 6 1 1350.2.j.a 4
45.h odd 6 1 4050.2.c.r 2
45.j even 6 1 inner 450.2.j.e 4
45.j even 6 1 4050.2.c.c 2
45.k odd 12 1 18.2.c.a ✓ 2
45.k odd 12 1 162.2.a.c 1
45.k odd 12 1 450.2.e.i 2
45.k odd 12 1 4050.2.a.c 1
45.l even 12 1 54.2.c.a 2
45.l even 12 1 162.2.a.b 1
45.l even 12 1 1350.2.e.c 2
45.l even 12 1 4050.2.a.v 1
60.l odd 4 1 432.2.i.b 2
105.k odd 4 1 2646.2.f.g 2
105.w odd 12 1 2646.2.e.c 2
105.w odd 12 1 2646.2.h.i 2
105.x even 12 1 2646.2.e.b 2
105.x even 12 1 2646.2.h.h 2
120.q odd 4 1 1728.2.i.f 2
120.w even 4 1 1728.2.i.e 2
180.v odd 12 1 432.2.i.b 2
180.v odd 12 1 1296.2.a.f 1
180.x even 12 1 144.2.i.c 2
180.x even 12 1 1296.2.a.g 1
315.bs even 12 1 882.2.h.b 2
315.bt odd 12 1 882.2.h.c 2
315.bu odd 12 1 2646.2.h.i 2
315.bv even 12 1 2646.2.h.h 2
315.bw odd 12 1 2646.2.e.c 2
315.bx even 12 1 2646.2.e.b 2
315.cb even 12 1 882.2.f.d 2
315.cb even 12 1 7938.2.a.x 1
315.cf odd 12 1 2646.2.f.g 2
315.cf odd 12 1 7938.2.a.i 1
315.cg even 12 1 882.2.e.g 2
315.ch odd 12 1 882.2.e.i 2
360.bo even 12 1 576.2.i.a 2
360.bo even 12 1 5184.2.a.o 1
360.br even 12 1 1728.2.i.e 2
360.br even 12 1 5184.2.a.q 1
360.bt odd 12 1 1728.2.i.f 2
360.bt odd 12 1 5184.2.a.p 1
360.bu odd 12 1 576.2.i.g 2
360.bu odd 12 1 5184.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a ✓ 2 5.c odd 4 1
18.2.c.a ✓ 2 45.k odd 12 1
54.2.c.a 2 15.e even 4 1
54.2.c.a 2 45.l even 12 1
144.2.i.c 2 20.e even 4 1
144.2.i.c 2 180.x even 12 1
162.2.a.b 1 45.l even 12 1
162.2.a.c 1 45.k odd 12 1
432.2.i.b 2 60.l odd 4 1
432.2.i.b 2 180.v odd 12 1
450.2.e.i 2 5.c odd 4 1
450.2.e.i 2 45.k odd 12 1
450.2.j.e 4 1.a even 1 1 trivial
450.2.j.e 4 5.b even 2 1 inner
450.2.j.e 4 9.c even 3 1 inner
450.2.j.e 4 45.j even 6 1 inner
576.2.i.a 2 40.k even 4 1
576.2.i.a 2 360.bo even 12 1
576.2.i.g 2 40.i odd 4 1
576.2.i.g 2 360.bu odd 12 1
882.2.e.g 2 35.k even 12 1
882.2.e.g 2 315.cg even 12 1
882.2.e.i 2 35.l odd 12 1
882.2.e.i 2 315.ch odd 12 1
882.2.f.d 2 35.f even 4 1
882.2.f.d 2 315.cb even 12 1
882.2.h.b 2 35.k even 12 1
882.2.h.b 2 315.bs even 12 1
882.2.h.c 2 35.l odd 12 1
882.2.h.c 2 315.bt odd 12 1
1296.2.a.f 1 180.v odd 12 1
1296.2.a.g 1 180.x even 12 1
1350.2.e.c 2 15.e even 4 1
1350.2.e.c 2 45.l even 12 1
1350.2.j.a 4 3.b odd 2 1
1350.2.j.a 4 9.d odd 6 1
1350.2.j.a 4 15.d odd 2 1
1350.2.j.a 4 45.h odd 6 1
1728.2.i.e 2 120.w even 4 1
1728.2.i.e 2 360.br even 12 1
1728.2.i.f 2 120.q odd 4 1
1728.2.i.f 2 360.bt odd 12 1
2646.2.e.b 2 105.x even 12 1
2646.2.e.b 2 315.bx even 12 1
2646.2.e.c 2 105.w odd 12 1
2646.2.e.c 2 315.bw odd 12 1
2646.2.f.g 2 105.k odd 4 1
2646.2.f.g 2 315.cf odd 12 1
2646.2.h.h 2 105.x even 12 1
2646.2.h.h 2 315.bv even 12 1
2646.2.h.i 2 105.w odd 12 1
2646.2.h.i 2 315.bu odd 12 1
4050.2.a.c 1 45.k odd 12 1
4050.2.a.v 1 45.l even 12 1
4050.2.c.c 2 9.c even 3 1
4050.2.c.c 2 45.j even 6 1
4050.2.c.r 2 9.d odd 6 1
4050.2.c.r 2 45.h odd 6 1
5184.2.a.o 1 360.bo even 12 1
5184.2.a.p 1 360.bt odd 12 1
5184.2.a.q 1 360.br even 12 1
5184.2.a.r 1 360.bu odd 12 1
7938.2.a.i 1 315.cf odd 12 1
7938.2.a.x 1 315.cb even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{4} - 4T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{19} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$71$ \( (T + 12)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
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