# Properties

 Label 361.3.d.a Level $361$ Weight $3$ Character orbit 361.d Analytic conductor $9.837$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.83653754341$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 \zeta_{6} q^{4} + ( 9 - 9 \zeta_{6} ) q^{5} -5 q^{7} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q -4 \zeta_{6} q^{4} + ( 9 - 9 \zeta_{6} ) q^{5} -5 q^{7} -9 \zeta_{6} q^{9} + 3 q^{11} + ( -16 + 16 \zeta_{6} ) q^{16} + ( -15 + 15 \zeta_{6} ) q^{17} -36 q^{20} + 30 \zeta_{6} q^{23} -56 \zeta_{6} q^{25} + 20 \zeta_{6} q^{28} + ( -45 + 45 \zeta_{6} ) q^{35} + ( -36 + 36 \zeta_{6} ) q^{36} + ( 85 - 85 \zeta_{6} ) q^{43} -12 \zeta_{6} q^{44} -81 q^{45} -75 \zeta_{6} q^{47} -24 q^{49} + ( 27 - 27 \zeta_{6} ) q^{55} -103 \zeta_{6} q^{61} + 45 \zeta_{6} q^{63} + 64 q^{64} + 60 q^{68} + ( 25 - 25 \zeta_{6} ) q^{73} -15 q^{77} + 144 \zeta_{6} q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} + 90 q^{83} + 135 \zeta_{6} q^{85} + ( 120 - 120 \zeta_{6} ) q^{92} -27 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 9 q^{5} - 10 q^{7} - 9 q^{9} + O(q^{10})$$ $$2 q - 4 q^{4} + 9 q^{5} - 10 q^{7} - 9 q^{9} + 6 q^{11} - 16 q^{16} - 15 q^{17} - 72 q^{20} + 30 q^{23} - 56 q^{25} + 20 q^{28} - 45 q^{35} - 36 q^{36} + 85 q^{43} - 12 q^{44} - 162 q^{45} - 75 q^{47} - 48 q^{49} + 27 q^{55} - 103 q^{61} + 45 q^{63} + 128 q^{64} + 120 q^{68} + 25 q^{73} - 30 q^{77} + 144 q^{80} - 81 q^{81} + 180 q^{83} + 135 q^{85} + 120 q^{92} - 27 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 −2.00000 + 3.46410i 4.50000 + 7.79423i 0 −5.00000 0 −4.50000 + 7.79423i 0
293.1 0 0 −2.00000 3.46410i 4.50000 7.79423i 0 −5.00000 0 −4.50000 7.79423i 0
 $$n$$: e.g. 2-40 or 990-1000 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 1 inner
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.3.d.a 2
19.b odd 2 1 CM 361.3.d.a 2
19.c even 3 1 19.3.b.a 1
19.c even 3 1 inner 361.3.d.a 2
19.d odd 6 1 19.3.b.a 1
19.d odd 6 1 inner 361.3.d.a 2
19.e even 9 6 361.3.f.a 6
19.f odd 18 6 361.3.f.a 6
57.f even 6 1 171.3.c.a 1
57.h odd 6 1 171.3.c.a 1
76.f even 6 1 304.3.e.a 1
76.g odd 6 1 304.3.e.a 1
95.h odd 6 1 475.3.c.a 1
95.i even 6 1 475.3.c.a 1
95.l even 12 2 475.3.d.a 2
95.m odd 12 2 475.3.d.a 2
152.k odd 6 1 1216.3.e.b 1
152.l odd 6 1 1216.3.e.a 1
152.o even 6 1 1216.3.e.b 1
152.p even 6 1 1216.3.e.a 1
228.m even 6 1 2736.3.o.a 1
228.n odd 6 1 2736.3.o.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 19.c even 3 1
19.3.b.a 1 19.d odd 6 1
171.3.c.a 1 57.f even 6 1
171.3.c.a 1 57.h odd 6 1
304.3.e.a 1 76.f even 6 1
304.3.e.a 1 76.g odd 6 1
361.3.d.a 2 1.a even 1 1 trivial
361.3.d.a 2 19.b odd 2 1 CM
361.3.d.a 2 19.c even 3 1 inner
361.3.d.a 2 19.d odd 6 1 inner
361.3.f.a 6 19.e even 9 6
361.3.f.a 6 19.f odd 18 6
475.3.c.a 1 95.h odd 6 1
475.3.c.a 1 95.i even 6 1
475.3.d.a 2 95.l even 12 2
475.3.d.a 2 95.m odd 12 2
1216.3.e.a 1 152.l odd 6 1
1216.3.e.a 1 152.p even 6 1
1216.3.e.b 1 152.k odd 6 1
1216.3.e.b 1 152.o even 6 1
2736.3.o.a 1 228.m even 6 1
2736.3.o.a 1 228.n odd 6 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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$81 - 9 T + T^{2}$$
$7$ $$( 5 + T )^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$225 + 15 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$900 - 30 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$7225 - 85 T + T^{2}$$
$47$ $$5625 + 75 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$10609 + 103 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$625 - 25 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -90 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$