Properties

Label 361.3.f.a
Level $361$
Weight $3$
Character orbit 361.f
Analytic conductor $9.837$
Analytic rank $0$
Dimension $6$
CM discriminant -19
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [361,3,Mod(116,361)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(361, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("361.116");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 361.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.83653754341\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
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: $\mathrm{U}(1)[D_{18}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q - 4 \zeta_{18} q^{4} + ( - 9 \zeta_{18}^{5} + 9 \zeta_{18}^{2}) q^{5} + 5 \zeta_{18}^{3} q^{7} + 9 \zeta_{18}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 \zeta_{18} q^{4} + ( - 9 \zeta_{18}^{5} + 9 \zeta_{18}^{2}) q^{5} + 5 \zeta_{18}^{3} q^{7} + 9 \zeta_{18}^{4} q^{9} + (3 \zeta_{18}^{3} - 3) q^{11} + 16 \zeta_{18}^{2} q^{16} - 15 \zeta_{18}^{5} q^{17} - 36 q^{20} + 30 \zeta_{18} q^{23} + ( - 56 \zeta_{18}^{4} + 56 \zeta_{18}) q^{25} - 20 \zeta_{18}^{4} q^{28} + 45 \zeta_{18}^{2} q^{35} - 36 \zeta_{18}^{5} q^{36} + ( - 85 \zeta_{18}^{5} + 85 \zeta_{18}^{2}) q^{43} + ( - 12 \zeta_{18}^{4} + 12 \zeta_{18}) q^{44} + 81 \zeta_{18}^{3} q^{45} + 75 \zeta_{18}^{4} q^{47} + ( - 24 \zeta_{18}^{3} + 24) q^{49} + 27 \zeta_{18}^{5} q^{55} - 103 \zeta_{18} q^{61} + (45 \zeta_{18}^{4} - 45 \zeta_{18}) q^{63} - 64 \zeta_{18}^{3} q^{64} + (60 \zeta_{18}^{3} - 60) q^{68} - 25 \zeta_{18}^{2} q^{73} - 15 q^{77} + 144 \zeta_{18} q^{80} + (81 \zeta_{18}^{5} - 81 \zeta_{18}^{2}) q^{81} - 90 \zeta_{18}^{3} q^{83} - 135 \zeta_{18}^{4} q^{85} - 120 \zeta_{18}^{2} q^{92} - 27 \zeta_{18} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 15 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 15 q^{7} - 9 q^{11} - 216 q^{20} + 243 q^{45} + 72 q^{49} - 192 q^{64} - 180 q^{68} - 90 q^{77} - 270 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{18}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
0.939693 + 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
−0.766044 + 0.642788i
−0.173648 0.984808i
0.939693 0.342020i
0 0 −3.75877 1.36808i 8.45723 3.07818i 0 2.50000 + 4.33013i 0 1.56283 + 8.86327i 0
127.1 0 0 0.694593 3.93923i −1.56283 8.86327i 0 2.50000 4.33013i 0 6.89440 + 5.78509i 0
262.1 0 0 3.06418 + 2.57115i −6.89440 + 5.78509i 0 2.50000 4.33013i 0 −8.45723 + 3.07818i 0
299.1 0 0 3.06418 2.57115i −6.89440 5.78509i 0 2.50000 + 4.33013i 0 −8.45723 3.07818i 0
307.1 0 0 0.694593 + 3.93923i −1.56283 + 8.86327i 0 2.50000 + 4.33013i 0 6.89440 5.78509i 0
333.1 0 0 −3.75877 + 1.36808i 8.45723 + 3.07818i 0 2.50000 4.33013i 0 1.56283 8.86327i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
19.c even 3 2 inner
19.d odd 6 2 inner
19.e even 9 3 inner
19.f odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.3.f.a 6
19.b odd 2 1 CM 361.3.f.a 6
19.c even 3 2 inner 361.3.f.a 6
19.d odd 6 2 inner 361.3.f.a 6
19.e even 9 1 19.3.b.a 1
19.e even 9 2 361.3.d.a 2
19.e even 9 3 inner 361.3.f.a 6
19.f odd 18 1 19.3.b.a 1
19.f odd 18 2 361.3.d.a 2
19.f odd 18 3 inner 361.3.f.a 6
57.j even 18 1 171.3.c.a 1
57.l odd 18 1 171.3.c.a 1
76.k even 18 1 304.3.e.a 1
76.l odd 18 1 304.3.e.a 1
95.o odd 18 1 475.3.c.a 1
95.p even 18 1 475.3.c.a 1
95.q odd 36 2 475.3.d.a 2
95.r even 36 2 475.3.d.a 2
152.s odd 18 1 1216.3.e.a 1
152.t even 18 1 1216.3.e.a 1
152.u odd 18 1 1216.3.e.b 1
152.v even 18 1 1216.3.e.b 1
228.u odd 18 1 2736.3.o.a 1
228.v even 18 1 2736.3.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 19.e even 9 1
19.3.b.a 1 19.f odd 18 1
171.3.c.a 1 57.j even 18 1
171.3.c.a 1 57.l odd 18 1
304.3.e.a 1 76.k even 18 1
304.3.e.a 1 76.l odd 18 1
361.3.d.a 2 19.e even 9 2
361.3.d.a 2 19.f odd 18 2
361.3.f.a 6 1.a even 1 1 trivial
361.3.f.a 6 19.b odd 2 1 CM
361.3.f.a 6 19.c even 3 2 inner
361.3.f.a 6 19.d odd 6 2 inner
361.3.f.a 6 19.e even 9 3 inner
361.3.f.a 6 19.f odd 18 3 inner
475.3.c.a 1 95.o odd 18 1
475.3.c.a 1 95.p even 18 1
475.3.d.a 2 95.q odd 36 2
475.3.d.a 2 95.r even 36 2
1216.3.e.a 1 152.s odd 18 1
1216.3.e.a 1 152.t even 18 1
1216.3.e.b 1 152.u odd 18 1
1216.3.e.b 1 152.v even 18 1
2736.3.o.a 1 228.u odd 18 1
2736.3.o.a 1 228.v even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(361, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 729 T^{3} + 531441 \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 25)^{3} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 3375 T^{3} + \cdots + 11390625 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 27000 T^{3} + \cdots + 729000000 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - 614125 T^{3} + \cdots + 377149515625 \) Copy content Toggle raw display
$47$ \( T^{6} + 421875 T^{3} + \cdots + 177978515625 \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 1092727 T^{3} + \cdots + 1194052296529 \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - 15625 T^{3} + \cdots + 244140625 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( (T^{2} + 90 T + 8100)^{3} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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