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 $12$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 361.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.83653754341\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
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

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.

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

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.

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 333.1
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 \(S_{3}^{\mathrm{new}}(361, [\chi])\).

Hecke characteristic polynomials

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