Newform orbit 300.2.e.c

• Label: 300.2.e.c
• Level: $300$
• Weight: $2$
• Character orbit: 300.e
• Analytic conductor: $2.396$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	300.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.39551206064\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.342102016.5
Defining polynomial:	\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b5 * q^2 + (b7 - b6) * q^3 + (b7 + b4 + b3) * q^4 + (-b4 - b3 - b2 + b1 - 1) * q^6 + (b4 + b3) * q^7 + (-b5 + b4 - b3) * q^8 + (b7 + b5 + b4 + b2) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{6} - 3\nu^{4} + 8\nu^{3} - 10\nu^{2} - 8\nu - 8 ) / 16 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} - \nu^{6} + 3\nu^{5} - 3\nu^{4} + 2\nu^{3} + 6\nu^{2} - 8 ) / 16 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{6} + 4\nu^{5} - 3\nu^{4} + 4\nu^{3} - 10\nu^{2} + 16\nu - 8 ) / 16 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{6} - 4\nu^{5} - 3\nu^{4} - 4\nu^{3} - 10\nu^{2} - 16\nu - 8 ) / 16 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} - \nu^{5} - 4\nu^{3} - 4\nu ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{7} - \nu^{6} - 3\nu^{5} - 3\nu^{4} - 2\nu^{3} + 6\nu^{2} - 8 ) / 16 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{6} + \nu^{4} + 2\nu^{2} ) / 8 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

251.1

−1.17915	+	0.780776i
−1.17915	−	0.780776i
−0.599676	+	1.28078i
−0.599676	−	1.28078i
0.599676	+	1.28078i
0.599676	−	1.28078i
1.17915	+	0.780776i
1.17915	−	0.780776i

−1.17915

−

0.780776i

1.51022

−

0.848071i

0.780776

+

1.84130i

0

−2.44293

−

0.179147i

3.02045i

0.516994

−

2.78078i

1.56155

−

2.56155i

0

251.2

−1.17915

+

0.780776i

1.51022

+

0.848071i

0.780776

−

1.84130i

0

−2.44293

+

0.179147i

−

3.02045i

0.516994

+

2.78078i

1.56155

+

2.56155i

0

251.3

−0.599676

−

1.28078i

−0.468213

+

1.66757i

−1.28078

+

1.53610i

0

2.41656

−

0.400324i

0.936426i

2.73546

+

0.719224i

−2.56155

−

1.56155i

0

251.4

−0.599676

+

1.28078i

−0.468213

−

1.66757i

−1.28078

−

1.53610i

0

2.41656

+

0.400324i

−

0.936426i

2.73546

−

0.719224i

−2.56155

+

1.56155i

0

251.5

0.599676

−

1.28078i

0.468213

−

1.66757i

−1.28078

−

1.53610i

0

−1.85500

−

1.59968i

−

0.936426i

−2.73546

+

0.719224i

−2.56155

−

1.56155i

0

251.6

0.599676

+

1.28078i

0.468213

+

1.66757i

−1.28078

+

1.53610i

0

−1.85500

+

1.59968i

0.936426i

−2.73546

−

0.719224i

−2.56155

+

1.56155i

0

251.7

1.17915

−

0.780776i

−1.51022

+

0.848071i

0.780776

−

1.84130i

0

−1.11862

+

2.17915i

−

3.02045i

−0.516994

−

2.78078i

1.56155

−

2.56155i

0

251.8

1.17915

+

0.780776i

−1.51022

−

0.848071i

0.780776

+

1.84130i

0

−1.11862

−

2.17915i

3.02045i

−0.516994

+

2.78078i

1.56155

+

2.56155i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	300.2.e.c		8
3.b	odd	2	1	inner	300.2.e.c		8
4.b	odd	2	1	inner	300.2.e.c		8
5.b	even	2	1		60.2.e.a	✓	8
5.c	odd	4	1		300.2.h.a		8
5.c	odd	4	1		300.2.h.b		8
12.b	even	2	1	inner	300.2.e.c		8
15.d	odd	2	1		60.2.e.a	✓	8
15.e	even	4	1		300.2.h.a		8
15.e	even	4	1		300.2.h.b		8
20.d	odd	2	1		60.2.e.a	✓	8
20.e	even	4	1		300.2.h.a		8
20.e	even	4	1		300.2.h.b		8
40.e	odd	2	1		960.2.h.g		8
40.f	even	2	1		960.2.h.g		8
60.h	even	2	1		60.2.e.a	✓	8
60.l	odd	4	1		300.2.h.a		8
60.l	odd	4	1		300.2.h.b		8
120.i	odd	2	1		960.2.h.g		8
120.m	even	2	1		960.2.h.g		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.2.e.a	✓	8	5.b	even	2	1
60.2.e.a	✓	8	15.d	odd	2	1
60.2.e.a	✓	8	20.d	odd	2	1
60.2.e.a	✓	8	60.h	even	2	1
300.2.e.c		8	1.a	even	1	1	trivial
300.2.e.c		8	3.b	odd	2	1	inner
300.2.e.c		8	4.b	odd	2	1	inner
300.2.e.c		8	12.b	even	2	1	inner
300.2.h.a		8	5.c	odd	4	1
300.2.h.a		8	15.e	even	4	1
300.2.h.a		8	20.e	even	4	1
300.2.h.a		8	60.l	odd	4	1
300.2.h.b		8	5.c	odd	4	1
300.2.h.b		8	15.e	even	4	1
300.2.h.b		8	20.e	even	4	1
300.2.h.b		8	60.l	odd	4	1
960.2.h.g		8	40.e	odd	2	1
960.2.h.g		8	40.f	even	2	1
960.2.h.g		8	120.i	odd	2	1
960.2.h.g		8	120.m	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{4} + 10T_{7}^{2} + 8 \)

\( T_{13}^{2} - 2T_{13} - 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 + T^6 + 4*T^4 + 4*T^2 + 16
$3$	T^8 + 2*T^6 + 2*T^4 + 18*T^2 + 81
$5$	\( T^{8} \)
$7$	\( (T^{4} + 10 T^{2} + 8)^{2} \)
$11$	\( (T^{4} - 20 T^{2} + 32)^{2} \)