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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [361,2,Mod(28,361)] 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("361.28"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(361, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 361.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,3,0,0,0,-9,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.88259951297\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{18} q^{3} + 2 \zeta_{18}^{5} q^{4} + 3 \zeta_{18}^{4} q^{5} + ( - \zeta_{18}^{3} + 1) q^{7} + \zeta_{18}^{2} q^{9} - 3 \zeta_{18}^{3} q^{11} + ( - 4 \zeta_{18}^{3} + 4) q^{12} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{2}) q^{13} + \cdots - 3 \zeta_{18}^{5} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{7} - 9 q^{11} + 12 q^{12} - 36 q^{20} + 12 q^{27} - 12 q^{31} - 12 q^{37} + 48 q^{39} - 9 q^{45} + 18 q^{49} + 24 q^{64} - 36 q^{65} - 18 q^{68} + 48 q^{75} - 18 q^{77} - 36 q^{83} - 12 q^{84}+ \cdots + 36 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
−0.766044 + 0.642788i
0.939693 − 0.342020i
−0.173648 + 0.984808i
−0.173648 − 0.984808i
0.939693 + 0.342020i
−0.766044 − 0.642788i
0 1.53209 − 1.28558i 1.87939 − 0.684040i −2.81908 − 1.02606i 0 0.500000 − 0.866025i 0 0.173648 − 0.984808i 0
54.1 0 −1.87939 + 0.684040i −0.347296 − 1.96962i 0.520945 − 2.95442i 0 0.500000 + 0.866025i 0 0.766044 − 0.642788i 0
62.1 0 0.347296 − 1.96962i −1.53209 + 1.28558i 2.29813 + 1.92836i 0 0.500000 + 0.866025i 0 −0.939693 − 0.342020i 0
99.1 0 0.347296 + 1.96962i −1.53209 − 1.28558i 2.29813 − 1.92836i 0 0.500000 − 0.866025i 0 −0.939693 + 0.342020i 0
234.1 0 −1.87939 − 0.684040i −0.347296 + 1.96962i 0.520945 + 2.95442i 0 0.500000 − 0.866025i 0 0.766044 + 0.642788i 0
245.1 0 1.53209 + 1.28558i 1.87939 + 0.684040i −2.81908 + 1.02606i 0 0.500000 + 0.866025i 0 0.173648 + 0.984808i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 28.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 2 inner
19.e even 9 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.e.e 6
19.b odd 2 1 361.2.e.d 6
19.c even 3 2 inner 361.2.e.e 6
19.d odd 6 2 361.2.e.d 6
19.e even 9 1 361.2.a.b 1
19.e even 9 2 361.2.c.a 2
19.e even 9 3 inner 361.2.e.e 6
19.f odd 18 1 19.2.a.a ✓ 1
19.f odd 18 2 361.2.c.c 2
19.f odd 18 3 361.2.e.d 6
57.j even 18 1 171.2.a.b 1
57.l odd 18 1 3249.2.a.d 1
76.k even 18 1 304.2.a.f 1
76.l odd 18 1 5776.2.a.c 1
95.o odd 18 1 475.2.a.b 1
95.p even 18 1 9025.2.a.d 1
95.r even 36 2 475.2.b.a 2
133.ba even 18 1 931.2.a.a 1
133.bb even 18 1 931.2.f.b 2
133.bd odd 18 1 931.2.f.c 2
133.be odd 18 1 931.2.f.c 2
133.bf even 18 1 931.2.f.b 2
152.s odd 18 1 1216.2.a.o 1
152.v even 18 1 1216.2.a.b 1
209.p even 18 1 2299.2.a.b 1
228.u odd 18 1 2736.2.a.c 1
247.bl odd 18 1 3211.2.a.a 1
285.bf even 18 1 4275.2.a.i 1
323.t odd 18 1 5491.2.a.b 1
380.bb even 18 1 7600.2.a.c 1
399.bx odd 18 1 8379.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a ✓ 1 19.f odd 18 1
171.2.a.b 1 57.j even 18 1
304.2.a.f 1 76.k even 18 1
361.2.a.b 1 19.e even 9 1
361.2.c.a 2 19.e even 9 2
361.2.c.c 2 19.f odd 18 2
361.2.e.d 6 19.b odd 2 1
361.2.e.d 6 19.d odd 6 2
361.2.e.d 6 19.f odd 18 3
361.2.e.e 6 1.a even 1 1 trivial
361.2.e.e 6 19.c even 3 2 inner
361.2.e.e 6 19.e even 9 3 inner
475.2.a.b 1 95.o odd 18 1
475.2.b.a 2 95.r even 36 2
931.2.a.a 1 133.ba even 18 1
931.2.f.b 2 133.bb even 18 1
931.2.f.b 2 133.bf even 18 1
931.2.f.c 2 133.bd odd 18 1
931.2.f.c 2 133.be odd 18 1
1216.2.a.b 1 152.v even 18 1
1216.2.a.o 1 152.s odd 18 1
2299.2.a.b 1 209.p even 18 1
2736.2.a.c 1 228.u odd 18 1
3211.2.a.a 1 247.bl odd 18 1
3249.2.a.d 1 57.l odd 18 1
4275.2.a.i 1 285.bf even 18 1
5491.2.a.b 1 323.t odd 18 1
5776.2.a.c 1 76.l odd 18 1
7600.2.a.c 1 380.bb even 18 1
8379.2.a.j 1 399.bx odd 18 1
9025.2.a.d 1 95.p even 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(361, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{6} + 8T_{3}^{3} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 8T^{3} + 64 \) Copy content Toggle raw display
$5$ \( T^{6} + 27T^{3} + 729 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} + 64T^{3} + 4096 \) Copy content Toggle raw display
$17$ \( T^{6} - 27T^{3} + 729 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - 216 T^{3} + 46656 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{3} \) Copy content Toggle raw display
$37$ \( (T + 2)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 216 T^{3} + 46656 \) Copy content Toggle raw display
$43$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} - 27T^{3} + 729 \) Copy content Toggle raw display
$53$ \( T^{6} - 1728 T^{3} + 2985984 \) Copy content Toggle raw display
$59$ \( T^{6} + 216 T^{3} + 46656 \) Copy content Toggle raw display
$61$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$67$ \( T^{6} + 64T^{3} + 4096 \) Copy content Toggle raw display
$71$ \( T^{6} - 216 T^{3} + 46656 \) Copy content Toggle raw display
$73$ \( T^{6} - 343 T^{3} + 117649 \) Copy content Toggle raw display
$79$ \( T^{6} - 512 T^{3} + 262144 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 144)^{3} \) Copy content Toggle raw display
$89$ \( T^{6} - 1728 T^{3} + 2985984 \) Copy content Toggle raw display
$97$ \( T^{6} - 512 T^{3} + 262144 \) Copy content Toggle raw display
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