Newform orbit 735.2.i.m

• Label: 735.2.i.m
• Level: $735$
• Weight: $2$
• Character orbit: 735.i
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.1534132224.10
Defining polynomial:	\( x^{8} + 6x^{6} - 8x^{5} + 34x^{4} - 24x^{3} + 28x^{2} + 8x + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} - 8x^{5} + 34x^{4} - 24x^{3} + 28x^{2} + 8x + 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( 19\nu^{7} - 29\nu^{6} + 348\nu^{5} - 76\nu^{4} + 2204\nu^{3} - 2378\nu^{2} + 5300\nu - 348 ) / 1442 \)
\(\beta_{3}\)	\(=\)	\( ( 51\nu^{7} + 36\nu^{6} + 289\nu^{5} - 204\nu^{4} + 1590\nu^{3} + 68\nu^{2} + 34\nu + 432 ) / 1442 \)
\(\beta_{4}\)	\(=\)	\( ( -81\nu^{7} - 142\nu^{6} - 459\nu^{5} + 324\nu^{4} - 744\nu^{3} - 108\nu^{2} - 54\nu + 2622 ) / 1442 \)
\(\beta_{5}\)	\(=\)	\( ( -108\nu^{7} + 51\nu^{6} - 612\nu^{5} + 1153\nu^{4} - 3876\nu^{3} + 4182\nu^{2} - 2956\nu - 830 ) / 1442 \)
\(\beta_{6}\)	\(=\)	\( ( -24\nu^{7} - 23\nu^{6} - 136\nu^{5} + 96\nu^{4} - 518\nu^{3} - 32\nu^{2} - 16\nu - 276 ) / 206 \)
\(\beta_{7}\)	\(=\)	\( ( 188\nu^{7} - 249\nu^{6} + 825\nu^{5} - 2915\nu^{4} + 7388\nu^{3} - 8882\nu^{2} + 4932\nu + 1338 ) / 1442 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(-1 - \beta_{5}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

226.1

0.874559	−	1.51478i
0.635665	−	1.10100i
−0.167452	+	0.290035i
−1.34277	+	2.32575i
0.874559	+	1.51478i
0.635665	+	1.10100i
−0.167452	−	0.290035i
−1.34277	−	2.32575i

−1.37456

+

2.38081i

−0.500000

−

0.866025i

−2.77882

−

4.81306i

0.500000

−

0.866025i

2.74912

0

9.78039

−0.500000

+

0.866025i

1.37456

+

2.38081i

226.2

−1.13567

+

1.96703i

−0.500000

−

0.866025i

−1.57947

−

2.73572i

0.500000

−

0.866025i

2.27133

0

2.63234

−0.500000

+

0.866025i

1.13567

+

1.96703i

226.3

−0.332548

+

0.575990i

−0.500000

−

0.866025i

0.778824

+

1.34896i

0.500000

−

0.866025i

0.665096

0

−2.36618

−0.500000

+

0.866025i

0.332548

+

0.575990i

226.4

0.842772

−

1.45972i

−0.500000

−

0.866025i

−0.420529

−

0.728378i

0.500000

−

0.866025i

−1.68554

0

1.95345

−0.500000

+

0.866025i

−0.842772

−

1.45972i

361.1

−1.37456

−

2.38081i

−0.500000

+

0.866025i

−2.77882

+

4.81306i

0.500000

+

0.866025i

2.74912

0

9.78039

−0.500000

−

0.866025i

1.37456

−

2.38081i

361.2

−1.13567

−

1.96703i

−0.500000

+

0.866025i

−1.57947

+

2.73572i

0.500000

+

0.866025i

2.27133

0

2.63234

−0.500000

−

0.866025i

1.13567

−

1.96703i

361.3

−0.332548

−

0.575990i

−0.500000

+

0.866025i

0.778824

−

1.34896i

0.500000

+

0.866025i

0.665096

0

−2.36618

−0.500000

−

0.866025i

0.332548

−

0.575990i

361.4

0.842772

+

1.45972i

−0.500000

+

0.866025i

−0.420529

+

0.728378i

0.500000

+

0.866025i

−1.68554

0

1.95345

−0.500000

−

0.866025i

−0.842772

+

1.45972i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.i.m		8
7.b	odd	2	1		735.2.i.n		8
7.c	even	3	1		735.2.a.o	yes	4
7.c	even	3	1	inner	735.2.i.m		8
7.d	odd	6	1		735.2.a.n	✓	4
7.d	odd	6	1		735.2.i.n		8
21.g	even	6	1		2205.2.a.bf		4
21.h	odd	6	1		2205.2.a.bg		4
35.i	odd	6	1		3675.2.a.bl		4
35.j	even	6	1		3675.2.a.bk		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.a.n	✓	4	7.d	odd	6	1
735.2.a.o	yes	4	7.c	even	3	1
735.2.i.m		8	1.a	even	1	1	trivial
735.2.i.m		8	7.c	even	3	1	inner
735.2.i.n		8	7.b	odd	2	1
735.2.i.n		8	7.d	odd	6	1
2205.2.a.bf		4	21.g	even	6	1
2205.2.a.bg		4	21.h	odd	6	1
3675.2.a.bk		4	35.j	even	6	1
3675.2.a.bl		4	35.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{8} + 4T_{2}^{7} + 16T_{2}^{6} + 24T_{2}^{5} + 55T_{2}^{4} + 56T_{2}^{3} + 144T_{2}^{2} + 84T_{2} + 49 \)

\( T_{13}^{4} - 32T_{13}^{2} - 64T_{13} + 16 \)

\( T_{17}^{8} + 8T_{17}^{7} + 80T_{17}^{6} + 128T_{17}^{5} + 1152T_{17}^{4} + 18432T_{17}^{2} - 16384T_{17} + 16384 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 + 4*T^7 + 16*T^6 + 24*T^5 + 55*T^4 + 56*T^3 + 144*T^2 + 84*T + 49
$3$	\( (T^{2} + T + 1)^{4} \)
$5$	\( (T^{2} - T + 1)^{4} \)
$7$	\( T^{8} \)
$11$	T^8 + 8*T^7 + 72*T^6 + 256*T^5 + 1568*T^4 + 4864*T^3 + 23808*T^2 + 35840*T + 50176