Properties

Label 300.3.g.c
Level 300300
Weight 33
Character orbit 300.g
Self dual yes
Analytic conductor 8.1748.174
Analytic rank 00
Dimension 11
CM discriminant -3
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [300,3,Mod(101,300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("300.101"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: N N == 300=22352 300 = 2^{2} \cdot 3 \cdot 5^{2}
Weight: k k == 3 3
Character orbit: [χ][\chi] == 300.g (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,0,0,13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 8.174407930818.17440793081
Analytic rank: 00
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{2}]

qq-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)
 
f(q)f(q) == q+3q3+13q7+9q923q13+11q19+39q21+27q27+59q3126q3769q3983q43+120q49+33q57121q61+117q63+13q67+46q73+167q97+O(q100) q + 3 q^{3} + 13 q^{7} + 9 q^{9} - 23 q^{13} + 11 q^{19} + 39 q^{21} + 27 q^{27} + 59 q^{31} - 26 q^{37} - 69 q^{39} - 83 q^{43} + 120 q^{49} + 33 q^{57} - 121 q^{61} + 117 q^{63} + 13 q^{67} + 46 q^{73}+ \cdots - 167 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/300Z)×\left(\mathbb{Z}/300\mathbb{Z}\right)^\times.

nn 101101 151151 277277
χ(n)\chi(n) 1-1 11 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
101.1
0
0 3.00000 0 0 0 13.0000 0 9.00000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by Q(3)\Q(\sqrt{-3})

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.g.c yes 1
3.b odd 2 1 CM 300.3.g.c yes 1
4.b odd 2 1 1200.3.l.a 1
5.b even 2 1 300.3.g.a 1
5.c odd 4 2 300.3.b.b 2
12.b even 2 1 1200.3.l.a 1
15.d odd 2 1 300.3.g.a 1
15.e even 4 2 300.3.b.b 2
20.d odd 2 1 1200.3.l.e 1
20.e even 4 2 1200.3.c.b 2
60.h even 2 1 1200.3.l.e 1
60.l odd 4 2 1200.3.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.b 2 5.c odd 4 2
300.3.b.b 2 15.e even 4 2
300.3.g.a 1 5.b even 2 1
300.3.g.a 1 15.d odd 2 1
300.3.g.c yes 1 1.a even 1 1 trivial
300.3.g.c yes 1 3.b odd 2 1 CM
1200.3.c.b 2 20.e even 4 2
1200.3.c.b 2 60.l odd 4 2
1200.3.l.a 1 4.b odd 2 1
1200.3.l.a 1 12.b even 2 1
1200.3.l.e 1 20.d odd 2 1
1200.3.l.e 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S3new(300,[χ])S_{3}^{\mathrm{new}}(300, [\chi]):

T713 T_{7} - 13 Copy content Toggle raw display
T11 T_{11} Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T T Copy content Toggle raw display
33 T3 T - 3 Copy content Toggle raw display
55 T T Copy content Toggle raw display
77 T13 T - 13 Copy content Toggle raw display
1111 T T Copy content Toggle raw display
1313 T+23 T + 23 Copy content Toggle raw display
1717 T T Copy content Toggle raw display
1919 T11 T - 11 Copy content Toggle raw display
2323 T T Copy content Toggle raw display
2929 T T Copy content Toggle raw display
3131 T59 T - 59 Copy content Toggle raw display
3737 T+26 T + 26 Copy content Toggle raw display
4141 T T Copy content Toggle raw display
4343 T+83 T + 83 Copy content Toggle raw display
4747 T T Copy content Toggle raw display
5353 T T Copy content Toggle raw display
5959 T T Copy content Toggle raw display
6161 T+121 T + 121 Copy content Toggle raw display
6767 T13 T - 13 Copy content Toggle raw display
7171 T T Copy content Toggle raw display
7373 T46 T - 46 Copy content Toggle raw display
7979 T+142 T + 142 Copy content Toggle raw display
8383 T T Copy content Toggle raw display
8989 T T Copy content Toggle raw display
9797 T+167 T + 167 Copy content Toggle raw display
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