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

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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

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