Newform orbit 405.4.e.g

• Label: 405.4.e.g
• Level: $405$
• Weight: $4$
• Character orbit: 405.e
• Analytic conductor: $23.896$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 15)
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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{2} + 7 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 24 \zeta_{6} + 24) q^{7} - 15 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 7 q^{4} - 5 q^{5} + 24 q^{7} - 30 q^{8}+O(q^{10}) \)

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

−0.500000

+

0.866025i

0

3.50000

+

6.06218i

−2.50000

−

4.33013i

0

12.0000

−

20.7846i

−15.0000

0

5.00000

271.1

−0.500000

−

0.866025i

0

3.50000

−

6.06218i

−2.50000

+

4.33013i

0

12.0000

+

20.7846i

−15.0000

0

5.00000

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.g		2
3.b	odd	2	1		405.4.e.i		2
9.c	even	3	1		15.4.a.a	✓	1
9.c	even	3	1	inner	405.4.e.g		2
9.d	odd	6	1		45.4.a.c		1
9.d	odd	6	1		405.4.e.i		2
36.f	odd	6	1		240.4.a.e		1
36.h	even	6	1		720.4.a.n		1
45.h	odd	6	1		225.4.a.f		1
45.j	even	6	1		75.4.a.b		1
45.k	odd	12	2		75.4.b.b		2
45.l	even	12	2		225.4.b.e		2
63.l	odd	6	1		735.4.a.e		1
63.o	even	6	1		2205.4.a.l		1
72.n	even	6	1		960.4.a.b		1
72.p	odd	6	1		960.4.a.ba		1
99.h	odd	6	1		1815.4.a.e		1
180.p	odd	6	1		1200.4.a.t		1
180.x	even	12	2		1200.4.f.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
15.4.a.a	✓	1	9.c	even	3	1
45.4.a.c		1	9.d	odd	6	1
75.4.a.b		1	45.j	even	6	1
75.4.b.b		2	45.k	odd	12	2
225.4.a.f		1	45.h	odd	6	1
225.4.b.e		2	45.l	even	12	2
240.4.a.e		1	36.f	odd	6	1
405.4.e.g		2	1.a	even	1	1	trivial
405.4.e.g		2	9.c	even	3	1	inner
405.4.e.i		2	3.b	odd	2	1
405.4.e.i		2	9.d	odd	6	1
720.4.a.n		1	36.h	even	6	1
735.4.a.e		1	63.l	odd	6	1
960.4.a.b		1	72.n	even	6	1
960.4.a.ba		1	72.p	odd	6	1
1200.4.a.t		1	180.p	odd	6	1
1200.4.f.b		2	180.x	even	12	2
1815.4.a.e		1	99.h	odd	6	1
2205.4.a.l		1	63.o	even	6	1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \)

\( T_{7}^{2} - 24T_{7} + 576 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 5T + 25 \)
$7$	\( T^{2} - 24T + 576 \)
$11$	\( T^{2} + 52T + 2704 \)