Newform orbit 735.2.g.a

• Label: 735.2.g.a
• Level: $735$
• Weight: $2$
• Character orbit: 735.g
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $16$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

Level:	$N$	$=$	$735 = 3 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	735.g (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.721389578983833600000000.5
Defining polynomial:	$x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{8}\cdot 7^{6}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 - b3 * q^2 + b5 * q^3 + (-b8 + 2) * q^4 + b11 * q^5 + b1 * q^6 + (b9 - 2*b3) * q^8 - 3 * q^9 + b6 * q^10 + (b14 + 2*b5) * q^12 + b10 * q^15 + (b10 - 2*b8 + 4) * q^16 - b4 * q^17 + 3*b3 * q^18 + b13 * q^19 + (-b14 + 2*b11 + b4) * q^20 + b7 * q^23 + (b13 - b6 + 2*b1) * q^24 - 5 * q^25 - 3*b5 * q^27 + (b9 - b7) * q^30 - b12 * q^31 + (b9 - b7 - 4*b3) * q^32 + (-b13 - b12 - b6 - b1) * q^34 + (3*b8 - 6) * q^36 + (b14 + b11 + b4) * q^38 + (b12 + 2*b6 - b1) * q^40 - 3*b11 * q^45 + (-3*b10 - b2) * q^46 + (-2*b14 + b4) * q^47 + (2*b14 - 3*b11 + 4*b5) * q^48 + 5*b3 * q^50 + (-b8 - b2) * q^51 + (-2*b9 + b7) * q^53 - 3*b1 * q^54 + (-2*b9 - b7) * q^57 + (2*b10 - 2*b8 + b2) * q^60 + (-b12 + 2*b1) * q^61 + (b14 + b5 - 2*b4) * q^62 + (2*b10 - 2*b8 + b2 + 8) * q^64 + (-5*b11 - 3*b5 - 2*b4) * q^68 + (b13 + 2*b6) * q^69 + (-3*b9 + 6*b3) * q^72 - 5*b5 * q^75 + (b12 + b6 + 3*b1) * q^76 + (-3*b8 + b2) * q^79 + (-2*b14 + 4*b11 - 5*b5 + 2*b4) * q^80 + 9 * q^81 + (-2*b14 - b4) * q^83 + (-3*b8 - b2) * q^85 - 3*b6 * q^90 + (-b15 - 3*b9 + 3*b7) * q^92 + (-b15 + b3) * q^93 + (-b13 + b12 + 3*b6 - 3*b1) * q^94 + (b15 + b3) * q^95 + (-3*b6 + 4*b1) * q^96
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 32 q^{4} - 48 q^{9} + 64 q^{16} - 80 q^{25} - 96 q^{36} + 128 q^{64} + 144 q^{81}+O(q^{100})$

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401$ :

$\beta_{1}$	$=$	(65*v^15 - 4078*v^13 - 51682*v^11 - 316357*v^9 - 1225031*v^7 - 2770459*v^5 - 1907888*v^3 + 4240068*v) / 4215813
$\beta_{2}$	$=$	(333*v^14 + 1814*v^12 + 20169*v^10 + 61722*v^8 + 226708*v^6 - 249174*v^4 - 1378692*v^2 - 6475154) / 743967
$\beta_{3}$	$=$	(467*v^15 + 3796*v^13 + 21270*v^11 + 62179*v^9 + 110885*v^7 - 225705*v^5 - 357836*v^3 - 366226*v) / 743967
$\beta_{4}$	$=$	(-416*v^14 - 5038*v^12 - 35507*v^10 - 164752*v^8 - 485725*v^6 - 905496*v^4 + 203815*v^2 + 1049286) / 602259
$\beta_{5}$	$=$	(3152*v^14 + 14536*v^12 + 57408*v^10 - 24278*v^8 - 497536*v^6 - 3376584*v^4 + 1229536*v^2 + 4619867) / 4215813
$\beta_{6}$	$=$	(-12788*v^15 - 118391*v^13 - 703899*v^11 - 2491939*v^9 - 6346228*v^7 - 4121229*v^5 + 1057316*v^3 - 11242315*v) / 12647439
$\beta_{7}$	$=$	(784*v^15 + 1690*v^13 + 2190*v^11 - 72529*v^9 - 255160*v^7 - 896622*v^5 + 2788235*v^3 + 3139430*v) / 743967
$\beta_{8}$	$=$	(753*v^14 + 6868*v^12 + 36990*v^10 + 108930*v^8 + 163589*v^6 - 387270*v^4 - 1297401*v^2 + 1299284) / 743967
$\beta_{9}$	$=$	(1036*v^15 + 6313*v^13 + 31659*v^11 + 62366*v^9 + 79892*v^7 - 710379*v^5 + 572429*v^3 + 669095*v) / 743967
$\beta_{10}$	$=$	$( 48\nu^{14} + 433\nu^{12} + 2308\nu^{10} + 7516\nu^{8} + 11488\nu^{6} - 26290\nu^{4} - 88652\nu^{2} + 52430 ) / 35427$
$\beta_{11}$	$=$	$( -80\nu^{14} - 587\nu^{12} - 3204\nu^{10} - 8992\nu^{8} - 18880\nu^{6} + 26304\nu^{4} - 15484\nu^{2} - 44590 ) / 52479$
$\beta_{12}$	$=$	(28079*v^15 + 207511*v^13 + 1032996*v^11 + 2276044*v^9 + 1085018*v^7 - 25896513*v^5 - 29876738*v^3 + 85821932*v) / 12647439
$\beta_{13}$	$=$	(-37616*v^15 - 315308*v^13 - 1717926*v^11 - 5390767*v^9 - 10004680*v^7 + 9998436*v^5 + 19314689*v^3 - 70216510*v) / 12647439
$\beta_{14}$	$=$	(-2311*v^14 - 15992*v^12 - 87844*v^10 - 230258*v^8 - 395405*v^6 + 1020426*v^4 - 518377*v^2 - 1649928) / 602259
$\beta_{15}$	$=$	(6422*v^15 + 41353*v^13 + 219822*v^11 + 507391*v^9 + 751901*v^7 - 3701286*v^5 + 729142*v^3 + 1073786*v) / 743967

Character values

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

$n$	$346$	$442$	$491$
$\chi(n)$	$-1$	$-1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

734.1

−1.36434	−	1.59774i
−1.36434	+	1.59774i
−1.22796	−	0.279124i
−1.22796	+	0.279124i
−0.701515	−	1.98043i
−0.701515	+	1.98043i
−0.372250	+	1.20300i
−0.372250	−	1.20300i
0.372250	−	1.20300i
0.372250	+	1.20300i
0.701515	+	1.98043i
0.701515	−	1.98043i
1.22796	+	0.279124i
1.22796	−	0.279124i
1.36434	+	1.59774i
1.36434	−	1.59774i

−2.72868

−

1.73205i

5.44572

2.23607i

4.72622i

0

−9.40228

−3.00000

−

6.10152i

734.2

−2.72868

1.73205i

5.44572

−

2.23607i

−

4.72622i

0

−9.40228

−3.00000

6.10152i

734.3

−2.45591

−

1.73205i

4.03151

−

2.23607i

4.25377i

0

−4.98920

−3.00000

5.49159i

734.4

−2.45591

1.73205i

4.03151

2.23607i

−

4.25377i

0

−4.98920

−3.00000

−

5.49159i

734.5

−1.40303

−

1.73205i

−0.0315060

−

2.23607i

2.43012i

0

2.85026

−3.00000

3.13727i

734.6

−1.40303

1.73205i

−0.0315060

2.23607i

−

2.43012i

0

2.85026

−3.00000

−

3.13727i

734.7

−0.744500

−

1.73205i

−1.44572

2.23607i

1.28951i

0

2.56534

−3.00000

−

1.66475i

734.8

−0.744500

1.73205i

−1.44572

−

2.23607i

−

1.28951i

0

2.56534

−3.00000

1.66475i

734.9

0.744500

−

1.73205i

−1.44572

2.23607i

−

1.28951i

0

−2.56534

−3.00000

1.66475i

734.10

0.744500

1.73205i

−1.44572

−

2.23607i

1.28951i

0

−2.56534

−3.00000

−

1.66475i

734.11

1.40303

−

1.73205i

−0.0315060

−

2.23607i

−

2.43012i

0

−2.85026

−3.00000

−

3.13727i

734.12

1.40303

1.73205i

−0.0315060

2.23607i

2.43012i

0

−2.85026

−3.00000

3.13727i

734.13

2.45591

−

1.73205i

4.03151

−

2.23607i

−

4.25377i

0

4.98920

−3.00000

−

5.49159i

734.14

2.45591

1.73205i

4.03151

2.23607i

4.25377i

0

4.98920

−3.00000

5.49159i

734.15

2.72868

−

1.73205i

5.44572

2.23607i

−

4.72622i

0

9.40228

−3.00000

6.10152i

734.16

2.72868

1.73205i

5.44572

−

2.23607i

4.72622i

0

9.40228

−3.00000

−

6.10152i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15})$
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner
35.c	odd	2	1	inner
105.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.g.a	✓	16
3.b	odd	2	1	inner	735.2.g.a	✓	16
5.b	even	2	1	inner	735.2.g.a	✓	16
7.b	odd	2	1	inner	735.2.g.a	✓	16
7.c	even	3	1		735.2.p.d		16
7.c	even	3	1		735.2.p.e		16
7.d	odd	6	1		735.2.p.d		16
7.d	odd	6	1		735.2.p.e		16
15.d	odd	2	1	CM	735.2.g.a	✓	16
21.c	even	2	1	inner	735.2.g.a	✓	16
21.g	even	6	1		735.2.p.d		16
21.g	even	6	1		735.2.p.e		16
21.h	odd	6	1		735.2.p.d		16
21.h	odd	6	1		735.2.p.e		16
35.c	odd	2	1	inner	735.2.g.a	✓	16
35.i	odd	6	1		735.2.p.d		16
35.i	odd	6	1		735.2.p.e		16
35.j	even	6	1		735.2.p.d		16
35.j	even	6	1		735.2.p.e		16
105.g	even	2	1	inner	735.2.g.a	✓	16
105.o	odd	6	1		735.2.p.d		16
105.o	odd	6	1		735.2.p.e		16
105.p	even	6	1		735.2.p.d		16
105.p	even	6	1		735.2.p.e		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.g.a	✓	16	1.a	even	1	1	trivial
735.2.g.a	✓	16	3.b	odd	2	1	inner
735.2.g.a	✓	16	5.b	even	2	1	inner
735.2.g.a	✓	16	7.b	odd	2	1	inner
735.2.g.a	✓	16	15.d	odd	2	1	CM
735.2.g.a	✓	16	21.c	even	2	1	inner
735.2.g.a	✓	16	35.c	odd	2	1	inner
735.2.g.a	✓	16	105.g	even	2	1	inner
735.2.p.d		16	7.c	even	3	1
735.2.p.d		16	7.d	odd	6	1
735.2.p.d		16	21.g	even	6	1
735.2.p.d		16	21.h	odd	6	1
735.2.p.d		16	35.i	odd	6	1
735.2.p.d		16	35.j	even	6	1
735.2.p.d		16	105.o	odd	6	1
735.2.p.d		16	105.p	even	6	1
735.2.p.e		16	7.c	even	3	1
735.2.p.e		16	7.d	odd	6	1
735.2.p.e		16	21.g	even	6	1
735.2.p.e		16	21.h	odd	6	1
735.2.p.e		16	35.i	odd	6	1
735.2.p.e		16	35.j	even	6	1
735.2.p.e		16	105.o	odd	6	1
735.2.p.e		16	105.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{8} - 16T_{2}^{6} + 80T_{2}^{4} - 128T_{2}^{2} + 49$ acting on $S_{2}^{\mathrm{new}}(735, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^8 - 16*T^6 + 80*T^4 - 128*T^2 + 49)^2
$3$	$(T^{2} + 3)^{8}$
$5$	$(T^{2} + 5)^{8}$
$7$	$T^{16}$
$11$	$T^{16}$