Newform orbit 605.2.m.a

• Label: 605.2.m.a
• Level: $605$
• Weight: $2$
• Character orbit: 605.m
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $16$
• CM discriminant: -11
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.m (of order \(20\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{20})\)
Coefficient field:	16.0.3429742096000000000000.1
Defining polynomial:	\( x^{16} + 5x^{14} + 16x^{12} + 35x^{10} + 31x^{8} + 315x^{6} + 1296x^{4} + 3645x^{2} + 6561 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b5 + b4 + b1) * q^3 + (2*b14 + 2*b10 - 2*b7 - 2*b5) * q^4 + (-b12 + 2*b10) * q^5 + (3*b14 + 2*b8 + b3) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 5x^{14} + 16x^{12} + 35x^{10} + 31x^{8} + 315x^{6} + 1296x^{4} + 3645x^{2} + 6561 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{14} + 718\nu^{4} ) / 2511 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{12} - 160\nu^{2} ) / 279 \)
\(\beta_{4}\)	\(=\)	\( ( -5\nu^{14} - 16\nu^{12} - 35\nu^{10} - 31\nu^{8} + 403\nu^{6} - 1296\nu^{4} - 3645\nu^{2} - 6561 ) / 7533 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{11} - 253\nu ) / 93 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{10} - 253 ) / 31 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{15} - 718\nu^{5} ) / 7533 \)
\(\beta_{8}\)	\(=\)	\( ( -2\nu^{12} - 599\nu^{2} ) / 279 \)
\(\beta_{9}\)	\(=\)	\( ( 35\nu^{14} + 175\nu^{12} + 560\nu^{10} + 496\nu^{8} + 1085\nu^{6} + 11025\nu^{4} + 45360\nu^{2} + 127575 ) / 22599 \)
\(\beta_{10}\)	\(=\)	\( ( 2\nu^{13} + 599\nu^{3} ) / 837 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{15} + 5\nu^{13} + 16\nu^{11} + 35\nu^{9} + 31\nu^{7} + 315\nu^{5} + 1296\nu^{3} + 3645\nu ) / 2187 \)
\(\beta_{12}\)	\(=\)	\( ( -\nu^{13} - 160\nu^{3} ) / 279 \)
\(\beta_{13}\)	\(=\)	\( ( -5\nu^{14} - 1079\nu^{4} ) / 2511 \)
\(\beta_{14}\)	\(=\)	\( ( 65\nu^{15} + 208\nu^{13} + 455\nu^{11} + 403\nu^{9} + 2294\nu^{7} + 16848\nu^{5} + 47385\nu^{3} + 85293\nu ) / 67797 \)
\(\beta_{15}\)	\(=\)	\( ( -5\nu^{15} - 1079\nu^{5} ) / 2511 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(\beta_{7}\)	\(-\beta_{13}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

112.1

−1.04771	+	1.37924i
1.63550	−	0.570223i
0.0369185	+	1.73166i
−0.987975	−	1.42264i
−1.63550	+	0.570223i
1.04771	−	1.37924i
0.0369185	−	1.73166i
−0.987975	+	1.42264i
−0.0369185	+	1.73166i
0.987975	−	1.42264i
−1.63550	−	0.570223i
1.04771	+	1.37924i
−1.04771	−	1.37924i
1.63550	+	0.570223i
−0.0369185	−	1.73166i
0.987975	+	1.42264i

0

−0.256255

+

1.61793i

1.17557

−

1.61803i

0.914138

−

2.04067i

0

0

0

0.301128

+

0.0978424i

0

112.2

0

0.477487

−

3.01474i

1.17557

−

1.61803i

1.93903

+

1.11362i

0

0

0

−6.00747

−

1.95194i

0

118.1

0

−0.743682

+

1.45956i

−1.90211

−

0.618034i

2.22328

−

0.238794i

0

0

0

0.186107

+

0.256155i

0

118.2

0

1.38572

−

2.71963i

−1.90211

−

0.618034i

−0.459925

−

2.18826i

0

0

0

−3.71282

−

5.11026i

0

233.1

0

−1.61793

−

0.256255i

−1.17557

+

1.61803i

−1.93903

−

1.11362i

0

0

0

−0.301128

−

0.0978424i

0

233.2

0

3.01474

+

0.477487i

−1.17557

+

1.61803i

−0.914138

+

2.04067i

0

0

0

6.00747

+

1.95194i

0

282.1

0

−0.743682

−

1.45956i

−1.90211

+

0.618034i

2.22328

+

0.238794i

0

0

0

0.186107

−

0.256155i

0

282.2

0

1.38572

+

2.71963i

−1.90211

+

0.618034i

−0.459925

+

2.18826i

0

0

0

−3.71282

+

5.11026i

0

403.1

0

−2.71963

+

1.38572i

1.90211

−

0.618034i

−2.22328

−

0.238794i

0

0

0

3.71282

−

5.11026i

0

403.2

0

1.45956

−

0.743682i

1.90211

−

0.618034i

0.459925

−

2.18826i

0

0

0

−0.186107

+

0.256155i

0

457.1

0

−1.61793

+

0.256255i

−1.17557

−

1.61803i

−1.93903

+

1.11362i

0

0

0

−0.301128

+

0.0978424i

0

457.2

0

3.01474

−

0.477487i

−1.17557

−

1.61803i

−0.914138

−

2.04067i

0

0

0

6.00747

−

1.95194i

0

578.1

0

−0.256255

−

1.61793i

1.17557

+

1.61803i

0.914138

+

2.04067i

0

0

0

0.301128

−

0.0978424i

0

578.2

0

0.477487

+

3.01474i

1.17557

+

1.61803i

1.93903

−

1.11362i

0

0

0

−6.00747

+

1.95194i

0

602.1

0

−2.71963

−

1.38572i

1.90211

+

0.618034i

−2.22328

+

0.238794i

0

0

0

3.71282

+

5.11026i

0

602.2

0

1.45956

+

0.743682i

1.90211

+

0.618034i

0.459925

+

2.18826i

0

0

0

−0.186107

−

0.256155i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by \(\Q(\sqrt{-11}) \)
5.c	odd	4	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
55.e	even	4	1	inner
55.k	odd	20	3	inner
55.l	even	20	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	605.2.m.a		16
5.c	odd	4	1	inner	605.2.m.a		16
11.b	odd	2	1	CM	605.2.m.a		16
11.c	even	5	1		55.2.e.b	✓	4
11.c	even	5	3	inner	605.2.m.a		16
11.d	odd	10	1		55.2.e.b	✓	4
11.d	odd	10	3	inner	605.2.m.a		16
33.f	even	10	1		495.2.k.a		4
33.h	odd	10	1		495.2.k.a		4
44.g	even	10	1		880.2.bd.d		4
44.h	odd	10	1		880.2.bd.d		4
55.e	even	4	1	inner	605.2.m.a		16
55.h	odd	10	1		275.2.e.a		4
55.j	even	10	1		275.2.e.a		4
55.k	odd	20	1		55.2.e.b	✓	4
55.k	odd	20	1		275.2.e.a		4
55.k	odd	20	3	inner	605.2.m.a		16
55.l	even	20	1		55.2.e.b	✓	4
55.l	even	20	1		275.2.e.a		4
55.l	even	20	3	inner	605.2.m.a		16
165.u	odd	20	1		495.2.k.a		4
165.v	even	20	1		495.2.k.a		4
220.v	even	20	1		880.2.bd.d		4
220.w	odd	20	1		880.2.bd.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
55.2.e.b	✓	4	11.c	even	5	1
55.2.e.b	✓	4	11.d	odd	10	1
55.2.e.b	✓	4	55.k	odd	20	1
55.2.e.b	✓	4	55.l	even	20	1
275.2.e.a		4	55.h	odd	10	1
275.2.e.a		4	55.j	even	10	1
275.2.e.a		4	55.k	odd	20	1
275.2.e.a		4	55.l	even	20	1
495.2.k.a		4	33.f	even	10	1
495.2.k.a		4	33.h	odd	10	1
495.2.k.a		4	165.u	odd	20	1
495.2.k.a		4	165.v	even	20	1
605.2.m.a		16	1.a	even	1	1	trivial
605.2.m.a		16	5.c	odd	4	1	inner
605.2.m.a		16	11.b	odd	2	1	CM
605.2.m.a		16	11.c	even	5	3	inner
605.2.m.a		16	11.d	odd	10	3	inner
605.2.m.a		16	55.e	even	4	1	inner
605.2.m.a		16	55.k	odd	20	3	inner
605.2.m.a		16	55.l	even	20	3	inner
880.2.bd.d		4	44.g	even	10	1
880.2.bd.d		4	44.h	odd	10	1
880.2.bd.d		4	220.v	even	20	1
880.2.bd.d		4	220.w	odd	20	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{16} \)
$3$	T^16 - 2*T^15 + 2*T^14 + 10*T^13 - 69*T^12 - 170*T^11 + 528*T^10 - 1656*T^9 + 2281*T^8 + 8280*T^7 + 13200*T^6 + 21250*T^5 - 43125*T^4 - 31250*T^3 + 31250*T^2 + 156250*T + 390625
$5$	T^16 + T^14 - 24*T^12 - 49*T^10 + 551*T^8 - 1225*T^6 - 15000*T^4 + 15625*T^2 + 390625
$7$	\( T^{16} \)
$11$	\( T^{16} \)