Newform orbit 961.2.d.i

• Label: 961.2.d.i
• Level: $961$
• Weight: $2$
• Character orbit: 961.d
• Analytic conductor: $7.674$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	961.d (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.67362363425\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.64000000.2
Defining polynomial:	\( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 31)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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 + (b7 - b2) * q^2 + (-b6 - b4 - b3 - b2 - 1) * q^3 + (2*b7 + 2*b5 + b4 + 2*b3 + 2*b1) * q^4 + q^5 + (2*b5 - 3) * q^6 + (b7 + b5 + b4 + b3 + b1) * q^7 + (-3*b6 + b1) * q^8 - 2*b1 * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 8 q^{5} - 24 q^{6} - 2 q^{7} + 6 q^{8}+O(q^{10}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(\beta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

374.1

0.437016	−	1.34500i
−0.437016	+	1.34500i
0.437016	+	1.34500i
−0.437016	−	1.34500i
1.14412	+	0.831254i
−1.14412	−	0.831254i
1.14412	−	0.831254i
−1.14412	+	0.831254i

−0.335106

−

0.243469i

0.335106

−

0.243469i

−0.565015

−

1.73894i

1.00000

−0.171573

−0.127999

−

0.393941i

−0.490035

+

1.50817i

−0.874032

+

2.68999i

−0.335106

−

0.243469i

374.2

1.95314

+

1.41904i

−1.95314

+

1.41904i

1.18305

+

3.64105i

1.00000

−5.82843

0.746033

+

2.29605i

−1.36407

+

4.19817i

0.874032

−

2.68999i

1.95314

+

1.41904i

388.1

−0.335106

+

0.243469i

0.335106

+

0.243469i

−0.565015

+

1.73894i

1.00000

−0.171573

−0.127999

+

0.393941i

−0.490035

−

1.50817i

−0.874032

−

2.68999i

−0.335106

+

0.243469i

388.2

1.95314

−

1.41904i

−1.95314

−

1.41904i

1.18305

−

3.64105i

1.00000

−5.82843

0.746033

−

2.29605i

−1.36407

−

4.19817i

0.874032

+

2.68999i

1.95314

−

1.41904i

531.1

−0.746033

−

2.29605i

0.746033

−

2.29605i

−3.09726

+

2.25029i

1.00000

−5.82843

−1.95314

+

1.41904i

3.57117

+

2.59461i

−2.28825

−

1.66251i

−0.746033

−

2.29605i

531.2

0.127999

+

0.393941i

−0.127999

+

0.393941i

1.47923

−

1.07472i

1.00000

−0.171573

0.335106

−

0.243469i

1.28293

+

0.932102i

2.28825

+

1.66251i

0.127999

+

0.393941i

628.1

−0.746033

+

2.29605i

0.746033

+

2.29605i

−3.09726

−

2.25029i

1.00000

−5.82843

−1.95314

−

1.41904i

3.57117

−

2.59461i

−2.28825

+

1.66251i

−0.746033

+

2.29605i

628.2

0.127999

−

0.393941i

−0.127999

−

0.393941i

1.47923

+

1.07472i

1.00000

−0.171573

0.335106

+

0.243469i

1.28293

−

0.932102i

2.28825

−

1.66251i

0.127999

−

0.393941i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.d	even	5	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	961.2.d.i		8
31.b	odd	2	1		961.2.d.l		8
31.c	even	3	2		961.2.g.r		16
31.d	even	5	1		961.2.a.c		2
31.d	even	5	3	inner	961.2.d.i		8
31.e	odd	6	2		961.2.g.o		16
31.f	odd	10	1		961.2.a.a		2
31.f	odd	10	3		961.2.d.l		8
31.g	even	15	2		961.2.c.a		4
31.g	even	15	6		961.2.g.r		16
31.h	odd	30	2		31.2.c.a	✓	4
31.h	odd	30	6		961.2.g.o		16
93.k	even	10	1		8649.2.a.l		2
93.l	odd	10	1		8649.2.a.k		2
93.p	even	30	2		279.2.h.c		4
124.p	even	30	2		496.2.i.h		4
155.v	odd	30	2		775.2.e.e		4
155.x	even	60	4		775.2.o.d		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
31.2.c.a	✓	4	31.h	odd	30	2
279.2.h.c		4	93.p	even	30	2
496.2.i.h		4	124.p	even	30	2
775.2.e.e		4	155.v	odd	30	2
775.2.o.d		8	155.x	even	60	4
961.2.a.a		2	31.f	odd	10	1
961.2.a.c		2	31.d	even	5	1
961.2.c.a		4	31.g	even	15	2
961.2.d.i		8	1.a	even	1	1	trivial
961.2.d.i		8	31.d	even	5	3	inner
961.2.d.l		8	31.b	odd	2	1
961.2.d.l		8	31.f	odd	10	3
961.2.g.o		16	31.e	odd	6	2
961.2.g.o		16	31.h	odd	30	6
961.2.g.r		16	31.c	even	3	2
961.2.g.r		16	31.g	even	15	6
8649.2.a.k		2	93.l	odd	10	1
8649.2.a.l		2	93.k	even	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{8} - 2T_{2}^{7} + 5T_{2}^{6} - 12T_{2}^{5} + 29T_{2}^{4} + 12T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \)

\( T_{3}^{8} + 2T_{3}^{7} + 5T_{3}^{6} + 12T_{3}^{5} + 29T_{3}^{4} - 12T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 - 2*T^7 + 5*T^6 - 12*T^5 + 29*T^4 + 12*T^3 + 5*T^2 + 2*T + 1
$3$	T^8 + 2*T^7 + 5*T^6 + 12*T^5 + 29*T^4 - 12*T^3 + 5*T^2 - 2*T + 1
$5$	\( (T - 1)^{8} \)
$7$	T^8 + 2*T^7 + 5*T^6 + 12*T^5 + 29*T^4 - 12*T^3 + 5*T^2 - 2*T + 1
$11$	T^8 + 2*T^7 + 21*T^6 + 76*T^5 + 509*T^4 - 1292*T^3 + 6069*T^2 - 9826*T + 83521