Newform orbit 405.4.e.b

• Label: 405.4.e.b
• Level: $405$
• Weight: $4$
• Character orbit: 405.e
• Analytic conductor: $23.896$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.8957735523\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (5 \zeta_{6} - 5) q^{2} - 17 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( - 30 \zeta_{6} + 30) q^{7} + 45 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{2} - 17 q^{4} + 5 q^{5} + 30 q^{7} + 90 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\)	\(82\)	\(326\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

136.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−2.50000

+

4.33013i

0

−8.50000

−

14.7224i

2.50000

+

4.33013i

0

15.0000

−

25.9808i

45.0000

0

−25.0000

271.1

−2.50000

−

4.33013i

0

−8.50000

+

14.7224i

2.50000

−

4.33013i

0

15.0000

+

25.9808i

45.0000

0

−25.0000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	405.4.e.b		2
3.b	odd	2	1		405.4.e.n		2
9.c	even	3	1		45.4.a.e	yes	1
9.c	even	3	1	inner	405.4.e.b		2
9.d	odd	6	1		45.4.a.a	✓	1
9.d	odd	6	1		405.4.e.n		2
36.f	odd	6	1		720.4.a.o		1
36.h	even	6	1		720.4.a.bc		1
45.h	odd	6	1		225.4.a.h		1
45.j	even	6	1		225.4.a.a		1
45.k	odd	12	2		225.4.b.b		2
45.l	even	12	2		225.4.b.a		2
63.l	odd	6	1		2205.4.a.t		1
63.o	even	6	1		2205.4.a.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.4.a.a	✓	1	9.d	odd	6	1
45.4.a.e	yes	1	9.c	even	3	1
225.4.a.a		1	45.j	even	6	1
225.4.a.h		1	45.h	odd	6	1
225.4.b.a		2	45.l	even	12	2
225.4.b.b		2	45.k	odd	12	2
405.4.e.b		2	1.a	even	1	1	trivial
405.4.e.b		2	9.c	even	3	1	inner
405.4.e.n		2	3.b	odd	2	1
405.4.e.n		2	9.d	odd	6	1
720.4.a.o		1	36.f	odd	6	1
720.4.a.bc		1	36.h	even	6	1
2205.4.a.a		1	63.o	even	6	1
2205.4.a.t		1	63.l	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} + 5T_{2} + 25 \)

\( T_{7}^{2} - 30T_{7} + 900 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 5T + 25 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 5T + 25 \)
$7$	\( T^{2} - 30T + 900 \)
$11$	\( T^{2} + 50T + 2500 \)