Newform orbit 900.6.d.a

• Label: 900.6.d.a
• Level: $900$
• Weight: $6$
• Character orbit: 900.d
• Analytic conductor: $144.345$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(144.345437832\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 44*b * q^7 - 540 * q^11 - 209*b * q^13 + 297*b * q^17 - 836 * q^19 + 2052*b * q^23 - 594 * q^29 + 4256 * q^31 + 149*b * q^37 - 17226 * q^41 - 6050*b * q^43 - 648*b * q^47 + 9063 * q^49 - 9747*b * q^53 - 7668 * q^59 - 34738 * q^61 - 10906*b * q^67 + 46872 * q^71 + 33781*b * q^73 - 23760*b * q^77 + 76912 * q^79 - 33858*b * q^83 + 29754 * q^89 + 36784 * q^91 + 61199*b * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1080 q^{11} - 1672 q^{19} - 1188 q^{29} + 8512 q^{31} - 34452 q^{41} + 18126 q^{49} - 15336 q^{59} - 69476 q^{61} + 93744 q^{71} + 153824 q^{79} + 59508 q^{89} + 73568 q^{91}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(451\)	\(577\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

649.1

	−	1.00000i
		1.00000i

0

0

0

0

0

−

88.0000i

0

0

0

649.2

0

0

0

0

0

88.0000i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.6.d.a		2
3.b	odd	2	1		100.6.c.b		2
5.b	even	2	1	inner	900.6.d.a		2
5.c	odd	4	1		36.6.a.a		1
5.c	odd	4	1		900.6.a.h		1
12.b	even	2	1		400.6.c.f		2
15.d	odd	2	1		100.6.c.b		2
15.e	even	4	1		4.6.a.a	✓	1
15.e	even	4	1		100.6.a.b		1
20.e	even	4	1		144.6.a.c		1
40.i	odd	4	1		576.6.a.bc		1
40.k	even	4	1		576.6.a.bd		1
45.k	odd	12	2		324.6.e.d		2
45.l	even	12	2		324.6.e.a		2
60.h	even	2	1		400.6.c.f		2
60.l	odd	4	1		16.6.a.b		1
60.l	odd	4	1		400.6.a.d		1
105.k	odd	4	1		196.6.a.e		1
105.w	odd	12	2		196.6.e.d		2
105.x	even	12	2		196.6.e.g		2
120.q	odd	4	1		64.6.a.b		1
120.w	even	4	1		64.6.a.f		1
165.l	odd	4	1		484.6.a.a		1
195.j	odd	4	1		676.6.d.a		2
195.s	even	4	1		676.6.a.a		1
195.u	odd	4	1		676.6.d.a		2
240.z	odd	4	1		256.6.b.c		2
240.bb	even	4	1		256.6.b.g		2
240.bd	odd	4	1		256.6.b.c		2
240.bf	even	4	1		256.6.b.g		2
420.w	even	4	1		784.6.a.d		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.6.a.a	✓	1	15.e	even	4	1
16.6.a.b		1	60.l	odd	4	1
36.6.a.a		1	5.c	odd	4	1
64.6.a.b		1	120.q	odd	4	1
64.6.a.f		1	120.w	even	4	1
100.6.a.b		1	15.e	even	4	1
100.6.c.b		2	3.b	odd	2	1
100.6.c.b		2	15.d	odd	2	1
144.6.a.c		1	20.e	even	4	1
196.6.a.e		1	105.k	odd	4	1
196.6.e.d		2	105.w	odd	12	2
196.6.e.g		2	105.x	even	12	2
256.6.b.c		2	240.z	odd	4	1
256.6.b.c		2	240.bd	odd	4	1
256.6.b.g		2	240.bb	even	4	1
256.6.b.g		2	240.bf	even	4	1
324.6.e.a		2	45.l	even	12	2
324.6.e.d		2	45.k	odd	12	2
400.6.a.d		1	60.l	odd	4	1
400.6.c.f		2	12.b	even	2	1
400.6.c.f		2	60.h	even	2	1
484.6.a.a		1	165.l	odd	4	1
576.6.a.bc		1	40.i	odd	4	1
576.6.a.bd		1	40.k	even	4	1
676.6.a.a		1	195.s	even	4	1
676.6.d.a		2	195.j	odd	4	1
676.6.d.a		2	195.u	odd	4	1
784.6.a.d		1	420.w	even	4	1
900.6.a.h		1	5.c	odd	4	1
900.6.d.a		2	1.a	even	1	1	trivial
900.6.d.a		2	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + 7744 \)

\( T_{11} + 540 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 7744 \)
$11$	\( (T + 540)^{2} \)