LMFDB - Newform orbit 99.2.m.a

Newspace parameters

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	99.m (of order $15$, degree $8$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.790518980011$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}$	$-1 - \zeta_{15}^{5}$

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} $

4.1

0.669131	+	0.743145i
−0.104528	+	0.994522i
0.669131	−	0.743145i
−0.104528	−	0.994522i
0.913545	−	0.406737i
−0.978148	−	0.207912i
−0.978148	+	0.207912i
0.913545	+	0.406737i

0.273659

−

2.60369i

−1.28716

1.15897i

−4.74803

−

1.00922i

−0.338261

−

3.21834i

2.66535

3.66854i

−0.669131

0.743145i

−2.30902

7.10642i

0.313585

−

2.98357i

−8.47214

16.1

0.373619

0.0794152i

1.72256

0.181049i

−1.69381

−

0.754131i

1.20906

−

0.256993i

0.629204

0.204441i

0.104528

0.994522i

−1.19098

−

0.865300i

2.93444

0.623735i

0.472136

25.1

0.273659

2.60369i

−1.28716

−

1.15897i

−4.74803

1.00922i

−0.338261

3.21834i

2.66535

−

3.66854i

−0.669131

−

0.743145i

−2.30902

−

7.10642i

0.313585

2.98357i

−8.47214

31.1

0.373619

−

0.0794152i

1.72256

−

0.181049i

−1.69381

0.754131i

1.20906

0.256993i

0.629204

−

0.204441i

0.104528

−

0.994522i

−1.19098

0.865300i

2.93444

−

0.623735i

0.472136

49.1

−0.255585

0.283856i

0.704489

1.58231i

0.193806

1.84395i

−0.827091

−

0.918578i

−0.629204

−

0.204441i

−0.913545

−

0.406737i

−1.19098

−

0.865300i

−2.00739

2.22943i

0.472136

58.1

−2.39169

−

1.06485i

0.360114

−

1.69420i

3.24803

3.60730i

2.95630

−

1.31623i

−2.66535

3.66854i

0.978148

−

0.207912i

−2.30902

−

7.10642i

−2.74064

−

1.22021i

−8.47214

70.1

−2.39169

1.06485i

0.360114

1.69420i

3.24803

−

3.60730i

2.95630

1.31623i

−2.66535

−

3.66854i

0.978148

0.207912i

−2.30902

7.10642i

−2.74064

1.22021i

−8.47214

97.1

−0.255585

−

0.283856i

0.704489

−

1.58231i

0.193806

−

1.84395i

−0.827091

0.918578i

−0.629204

0.204441i

−0.913545

0.406737i

−1.19098

0.865300i

−2.00739

−

2.22943i

0.472136

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
11.c	even	5	1	inner
99.m	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	99.2.m.a	✓	8
3.b	odd	2	1		297.2.n.a		8
9.c	even	3	1	inner	99.2.m.a	✓	8
9.c	even	3	1		891.2.f.b		4
9.d	odd	6	1		297.2.n.a		8
9.d	odd	6	1		891.2.f.a		4
11.c	even	5	1	inner	99.2.m.a	✓	8
11.c	even	5	1		1089.2.e.g		4
11.d	odd	10	1		1089.2.e.d		4
33.h	odd	10	1		297.2.n.a		8
99.m	even	15	1	inner	99.2.m.a	✓	8
99.m	even	15	1		891.2.f.b		4
99.m	even	15	1		1089.2.e.g		4
99.m	even	15	1		9801.2.a.n		2
99.n	odd	30	1		297.2.n.a		8
99.n	odd	30	1		891.2.f.a		4
99.n	odd	30	1		9801.2.a.bb		2
99.o	odd	30	1		1089.2.e.d		4
99.o	odd	30	1		9801.2.a.bc		2
99.p	even	30	1		9801.2.a.m		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
99.2.m.a	✓	8	1.a	even	1	1	trivial
99.2.m.a	✓	8	9.c	even	3	1	inner
99.2.m.a	✓	8	11.c	even	5	1	inner
99.2.m.a	✓	8	99.m	even	15	1	inner
297.2.n.a		8	3.b	odd	2	1
297.2.n.a		8	9.d	odd	6	1
297.2.n.a		8	33.h	odd	10	1
297.2.n.a		8	99.n	odd	30	1
891.2.f.a		4	9.d	odd	6	1
891.2.f.a		4	99.n	odd	30	1
891.2.f.b		4	9.c	even	3	1
891.2.f.b		4	99.m	even	15	1
1089.2.e.d		4	11.d	odd	10	1
1089.2.e.d		4	99.o	odd	30	1
1089.2.e.g		4	11.c	even	5	1
1089.2.e.g		4	99.m	even	15	1
9801.2.a.m		2	99.p	even	30	1
9801.2.a.n		2	99.m	even	15	1
9801.2.a.bb		2	99.n	odd	30	1
9801.2.a.bc		2	99.o	odd	30	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 4T_{2}^{7} + 10T_{2}^{6} + 26T_{2}^{5} + 39T_{2}^{4} - 14T_{2}^{3} - 5T_{2}^{2} - T_{2} + 1 $ T2^8 + 4*T2^7 + 10*T2^6 + 26*T2^5 + 39*T2^4 - 14*T2^3 - 5*T2^2 - T2 + 1 acting on $S_{2}^{\mathrm{new}}(99, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 4 T^{7} + 10 T^{6} + 26 T^{5} + \cdots + 1 $ T^8 + 4T^7 + 10T^6 + 26T^5 + 39T^4 - 14T^3 - 5T^2 - T + 1
$3$	$ T^{8} - 3 T^{7} + 6 T^{6} - 9 T^{5} + \cdots + 81 $ T^8 - 3T^7 + 6T^6 - 9T^5 + 9T^4 - 27T^3 + 54T^2 - 81*T + 81
$5$	$ T^{8} - 6 T^{7} + 20 T^{6} - 64 T^{5} + \cdots + 256 $ T^8 - 6T^7 + 20T^6 - 64T^5 + 144T^4 - 64T^3 - 256T + 256
$7$	$ T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 $ T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$11$	$ T^{8} - T^{7} - 20 T^{6} + T^{5} + \cdots + 14641 $ T^8 - T^7 - 20T^6 + T^5 + 309T^4 + 11T^3 - 2420T^2 - 1331*T + 14641
$13$	$ T^{8} - 8 T^{7} - 128 T^{5} + \cdots + 65536 $ T^8 - 8T^7 - 128T^5 + 2304T^4 - 8192T^3 + 20480T^2 - 49152T + 65536
$17$	$ (T^{4} - 6 T^{3} + 36 T^{2} - 81 T + 81)^{2} $ (T^4 - 6T^3 + 36T^2 - 81*T + 81)^2
$19$	$ (T^{4} + T^{3} + T^{2} + T + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$23$	$ (T^{4} + 7 T^{3} + 38 T^{2} + 77 T + 121)^{2} $ (T^4 + 7T^3 + 38T^2 + 77*T + 121)^2
$29$	$ T^{8} + 9 T^{7} + 45 T^{6} + \cdots + 6561 $ T^8 + 9T^7 + 45T^6 + 216T^5 + 729T^4 + 486T^3 + 4374T + 6561
$31$	$ T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1 $ T^8 + 3T^7 + 5T^6 + 8T^5 + 9T^4 + 2T^3 + 2T + 1
$37$	$ (T^{4} - 12 T^{3} + 94 T^{2} - 403 T + 961)^{2} $ (T^4 - 12T^3 + 94T^2 - 403*T + 961)^2
$41$	$ T^{8} - 3 T^{7} - 100 T^{6} + \cdots + 707281 $ T^8 - 3T^7 - 100T^6 - 1313T^5 + 12519T^4 + 58783T^3 + 151380T^2 + 414613*T + 707281
$43$	$ (T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81)^{2} $ (T^4 - 3T^3 + 18T^2 + 27*T + 81)^2
$47$	$ T^{8} - 23 T^{7} + 280 T^{6} + \cdots + 25411681 $ T^8 - 23T^7 + 280T^6 - 3313T^5 + 29199T^4 - 68657T^3 + 201640T^2 - 6084487*T + 25411681
$53$	$ (T^{4} - 14 T^{3} + 96 T^{2} - 319 T + 841)^{2} $ (T^4 - 14T^3 + 96T^2 - 319*T + 841)^2
$59$	$ T^{8} - 3 T^{7} + 5 T^{6} - 8 T^{5} + \cdots + 1 $ T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$61$	$ T^{8} - 90 T^{6} + 1350 T^{5} + \cdots + 4100625 $ T^8 - 90T^6 + 1350T^5 + 6075T^4 - 60750T^3 + 273375T^2 - 1366875T + 4100625
$67$	$ (T^{2} + 6 T + 36)^{4} $ (T^2 + 6*T + 36)^4
$71$	$ (T^{4} - 21 T^{3} + 166 T^{2} + 164 T + 1681)^{2} $ (T^4 - 21T^3 + 166T^2 + 164*T + 1681)^2
$73$	$ (T^{4} + 4 T^{3} + 46 T^{2} - 11 T + 1)^{2} $ (T^4 + 4T^3 + 46T^2 - 11*T + 1)^2
$79$	$ T^{8} + 22 T^{7} + \cdots + 104060401 $ T^8 + 22T^7 + 180T^6 + 2042T^5 + 31109T^4 + 257348T^3 + 2295225T^2 + 23696923*T + 104060401
$83$	$ T^{8} + 17 T^{7} + \cdots + 373301041 $ T^8 + 17T^7 + 130T^6 + 757T^5 - 10581T^4 - 502207T^3 - 2125310T^2 + 18799333*T + 373301041
$89$	$ (T^{2} + 18 T + 76)^{4} $ (T^2 + 18*T + 76)^4
$97$	$ T^{8} - 6 T^{7} + 216 T^{5} + \cdots + 1679616 $ T^8 - 6T^7 + 216T^5 - 1296T^4 + 7776T^3 - 279936*T + 1679616
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	99.m (of order \(15\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{15}^{7} + \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + 1) q^{2} + ( - 2 \zeta_{15}^{6} - \zeta_{15}) q^{3} + ( - 3 \zeta_{15}^{7} - 3 \zeta_{15}) q^{4} + (2 \zeta_{15}^{5} - 2 \zeta_{15} + 2) q^{5} + (2 \zeta_{15}^{7} + \zeta_{15}^{6} - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3} + 4 \zeta_{15} - 3) q^{6} + (\zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1) q^{7} + (4 \zeta_{15}^{7} - 5 \zeta_{15}^{6} + 4 \zeta_{15}^{2} - 4) q^{8} - 3 \zeta_{15}^{2} q^{9} +O(q^{10}) \) q + (-z^7 + z^6 + z^5 - z^4 + z^3 - 2z^2 + 1) q^2 + (-2z^6 - z) q^3 + (-3z^7 - 3z) * q^4 + (2z^5 - 2z + 2) * q^5 + (2z^7 + z^6 - 4z^5 + 2z^4 - z^3 + 4z - 3) * q^6 + (z^7 - z^6 + z^4 - z^3 + z^2 - 1) * q^7 + (4z^7 - 5z^6 + 4z^2 - 4) q^8 - 3z^2 q^9	\( q + ( - \zeta_{15}^{7} + \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + 1) q^{2} + ( - 2 \zeta_{15}^{6} - \zeta_{15}) q^{3} + ( - 3 \zeta_{15}^{7} - 3 \zeta_{15}) q^{4} + (2 \zeta_{15}^{5} - 2 \zeta_{15} + 2) q^{5} + (2 \zeta_{15}^{7} + \zeta_{15}^{6} - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3} + 4 \zeta_{15} - 3) q^{6} + (\zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1) q^{7} + (4 \zeta_{15}^{7} - 5 \zeta_{15}^{6} + 4 \zeta_{15}^{2} - 4) q^{8} - 3 \zeta_{15}^{2} q^{9} + ( - 4 \zeta_{15}^{7} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{2} - 2) q^{10} + ( - 2 \zeta_{15}^{7} + 4 \zeta_{15}^{6} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - \zeta_{15}^{2} + 2 \zeta_{15} + 2) q^{11} + (3 \zeta_{15}^{7} + 3 \zeta_{15}^{5} - 3 \zeta_{15}^{4} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{2} - 3 \zeta_{15} + 3) q^{12} + ( - 4 \zeta_{15}^{7} + 4 \zeta_{15}^{5} - 4 \zeta_{15}^{4} + 4) q^{13} + (\zeta_{15}^{7} - \zeta_{15}^{5} + 2 \zeta_{15} - 1) q^{14} + (4 \zeta_{15}^{7} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{2} + 2 \zeta_{15}) q^{15} + (5 \zeta_{15}^{7} + 3 \zeta_{15}^{6} - 8 \zeta_{15}^{5} + 5 \zeta_{15}^{4} - 5 \zeta_{15}^{3} + 8 \zeta_{15} - 5) q^{16} + (3 \zeta_{15}^{6} + 3 \zeta_{15}^{3} + 3) q^{17} + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{4} - 3 \zeta_{15}) q^{18} + \zeta_{15}^{6} q^{19} + (6 \zeta_{15}^{7} - 6 \zeta_{15}^{6} - 6 \zeta_{15}^{5} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{3} + 12 \zeta_{15}^{2} + \cdots - 6) q^{20} + \cdots + ( - 6 \zeta_{15}^{7} + 6 \zeta_{15}^{5} - 9 \zeta_{15}^{4} - 12 \zeta_{15} + 6) q^{99} +O(q^{100}) \) q + (-z^7 + z^6 + z^5 - z^4 + z^3 - 2z^2 + 1) q^2 + (-2z^6 - z) q^3 + (-3z^7 - 3z) * q^4 + (2z^5 - 2z + 2) * q^5 + (2z^7 + z^6 - 4z^5 + 2z^4 - z^3 + 4z - 3) * q^6 + (z^7 - z^6 + z^4 - z^3 + z^2 - 1) * q^7 + (4z^7 - 5z^6 + 4z^2 - 4) q^8 - 3z^2 q^9 + (-4z^7 + 4z^3 - 4z^2 - 2) q^10 + (-2z^7 + 4z^6 - 2z^4 + 2z^3 - z^2 + 2z + 2) q^11 + (3z^7 + 3z^5 - 3z^4 - 3z^3 + 3z^2 - 3z + 3) * q^12 + (-4z^7 + 4z^5 - 4z^4 + 4) q^13 + (z^7 - z^5 + 2z - 1) q^14 + (4z^7 - 2z^6 + 2z^2 + 2z) * q^15 + (5z^7 + 3z^6 - 8z^5 + 5z^4 - 5z^3 + 8z - 5) * q^16 + (3z^6 + 3z^3 + 3) * q^17 + (-3z^7 + 3z^4 - 3z) q^18 + z^6 * q^19 + (6z^7 - 6z^6 - 6z^5 + 6z^4 - 6z^3 + 12z^2 - 6) * q^20 + (2z^5 + 1) q^21 + (-3z^7 + 6z^5 - 3z^4 - 7z + 6) * q^22 + (-3z^5 - z^4 - z - 3) q^23 + (-9z^7 + 8z^6 + 4z^5 - 4z^4 + 8z^3 - 10z^2 + 4) * q^24 + (-3z^6 + 4z^5 + 4z^2 - 3z) * q^25 + (-4z^7 + 4z^6 - 4z^2) q^26 + (6z^7 - 6z^5 + 6z^4 - 3z^3 + 6z - 6) q^27 + (3z^6 + 3) q^28 + (3z^7 + 3z^4 - 3z^3 + 3z - 3) * q^29 + (-4z^7 + 12z^6 - 4z^5 - 8z^2 + 6z + 4) q^30 + (-z^5 + z^4 - 1) * q^31 + (9z^5 - 3z^4 - 3z + 9) q^32 + (-4z^7 - 2z^5 - 2z^4 + z^3 + 2z^2 - 2z) q^33 + (-3z^7 + 3z^5 - 3z^4 + 3z^3 - 3z^2 - 3z + 3) * q^34 + (2z^7 - 2z^6 - 2z^3 + 2z^2) * q^35 + (9z^7 - 9z^6 + 9z^2 - 9) q^36 + (3z^7 - 3z^6 + 2z^3 + 3z^2 + 2) * q^37 + (-z^7 - z^6 + 2z^5 - z^4 + z^3 - 2z + 1) * q^38 + (-4z^7 - 4z^6 - 4z^4 - 4z^3 - 4) * q^39 + (10z^7 - 8z^4 + 10z) q^40 + (-4z^7 + 4z^5 - 9z + 4) q^41 + (-3z^7 + z^6 - z^5 + z^4 + z^3 - 2z^2 - 1) * q^42 + (3z^7 - 3z^5 + 3z^4 - 3z^3 + 3z^2 + 3z - 3) * q^43 + (-9z^7 - 3z^6 + 6z^3 - 9z^2 - 3) * q^44 + (-6z^7 + 6z^3 - 6z^2) q^45 + (5z^7 - 2z^6 - 2z^3 + 5z^2) * q^46 + (z^6 - 8z^5 - 8z^2 + z) * q^47 + (2z^7 - 8z^6 + 10z^4 - 5z^3 + 8z^2 - 6z - 5) * q^48 + (6z^7 - 6z^5 + 6z^4 + 6z - 6) * q^49 + (3z^7 - 10z^5 + 3z^4 + 10z - 10) * q^50 + (-3z^7 - 3z^4 + 6z^3 - 3z + 6) * q^51 + (-12z^6 - 12z) * q^52 + (4z^7 - 5z^6 - 5z^3 + 4z^2) * q^53 + (3z^7 + 6z^5 - 3z^4 - 3z^3 + 3z^2 - 3z + 9) * q^54 + (-6z^7 + 8z^6 + 2z^3 - 6z^2) * q^55 + (-4z^7 + 5z^5 - 4z^4 + 4z^3 - 4z^2 - 4z + 4) * q^56 + (z^7 + 2z^2) q^57 + (-3z^7 + 3z^5 - 3z + 3) q^58 + (z^7 + z^4 + z) * q^59 + (-12z^7 - 6z^6 + 24z^5 - 12z^4 + 6z^3 - 24z + 18) * q^60 + (9z^7 - 3z^6 - 6z^5 + 9z^4 - 9z^3 + 6z - 9) * q^61 + (3z^7 - 3z^6 - z^3 + 3z^2 - 1) q^62 + 3z q^63 + (-5z^7 + 6z^6 + 6z^3 - 5z^2) * q^64 + 8z^5 q^65 + (11z^7 - 6z^6 - 3z^4 - 3z^3 + 7z^2 + 3z - 3) * q^66 + (-6z^5 - 6) q^67 - 9z^7 q^68 + (2z^7 + 3z^6 - z^5 + z^2 - 3z - 2) q^69 + (-2z^7 + 6z^6 - 2z^4 + 2z^3 - 6z^2 + 4z + 2) * q^70 + (5z^6 - 6z^3 + 5) * q^71 + (3z^7 + 12z^6 - 15z^5 + 3z^4 - 3z^3 + 15z - 3) * q^72 + (-5z^7 + 5z^6 + 2z^3 - 5z^2 + 2) * q^73 + (11z^6 - 8z^5 - 8z^2 + 11z) * q^74 + (-5z^7 + 4z^6 + 8z^5 - 8z^4 + 4z^3 - 3z^2 + 8) * q^75 + (-3z^5 + 3z^4 + 3z - 3) q^76 + (2z^7 - 4z^5 + 2z^4 + z - 4) q^77 + (8z^7 - 4z^5 + 4z^4 - 8z^3 + 8z^2 + 4z - 4) * q^78 + (-9z^7 + 9z^6 + 9z^5 - 9z^4 + 9z^3 - 4z^2 + 9) * q^79 + (-16z^7 + 22z^6 - 16z^2 + 16) q^80 + 9z^4 q^81 + (-13z^6 + 9z^3 - 13) * q^82 + (-11z^7 - 2z^6 + 13z^5 - 11z^4 + 11z^3 - 13z + 11) * q^83 + (3z^7 - 6z^6 + 6z^2 - 3z) * q^84 - 6z q^85 + (3z^7 + 6z^5 - 6z^4 + 6) q^86 + (3z^7 + 6z^4 - 3z^3 + 3z^2 + 6z - 3) q^87 + (15z^7 - 12z^6 - 13z^5 + 15z^4 - 15z^3 + 18z^2 + 3z - 15) q^88 + (-2z^7 + 2z^3 - 2z^2 - 8) q^89 + (12z^7 - 12z^6 - 12z^5 + 12z^4 - 12z^3 + 18z^2 - 12) * q^90 + (4z^7 + 4z^2 - 4) * q^91 + (3z^7 + 6z^6 + 3z^4 - 3z^3 - 6z^2 + 9z - 3) * q^92 + (z^6 + z^5 - z + 2) * q^93 + (-z^7 + 10z^5 - z^4 - 10z + 10) * q^94 + (-2z^7 - 2z) * q^95 + (6z^7 - 9z^6 - 3z^5 + 3z^2 + 9z - 6) q^96 + 6z^2 q^97 + (6z^7 - 6z^3 + 6z^2 + 6) q^98 + (-6z^7 + 6z^5 - 9z^4 - 12z + 6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 3 q^{3} - 6 q^{4} + 6 q^{5} - q^{7} - 14 q^{8} - 3 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 + 3 * q^3 - 6 * q^4 + 6 * q^5 - q^7 - 14 * q^8 - 3 * q^9	\( 8 q - 4 q^{2} + 3 q^{3} - 6 q^{4} + 6 q^{5} - q^{7} - 14 q^{8} - 3 q^{9} - 32 q^{10} + q^{11} + 18 q^{12} + 8 q^{13} - q^{14} + 12 q^{15} + 14 q^{16} + 12 q^{17} - 3 q^{18} - 2 q^{19} + 24 q^{20} + 11 q^{22} - 14 q^{23} - 39 q^{24} - 9 q^{25} - 16 q^{26} + 18 q^{28} - 9 q^{29} + 18 q^{30} - 3 q^{31} + 30 q^{32} - 6 q^{34} + 12 q^{35} - 36 q^{36} + 24 q^{37} - 4 q^{38} - 24 q^{39} + 12 q^{40} + 3 q^{41} - 12 q^{42} + 6 q^{43} - 48 q^{44} - 24 q^{45} + 18 q^{46} + 23 q^{47} - 6 q^{49} - 24 q^{50} + 27 q^{51} + 12 q^{52} + 28 q^{53} + 54 q^{54} - 32 q^{55} - 12 q^{56} + 3 q^{57} + 6 q^{58} + 3 q^{59} + 6 q^{62} + 3 q^{63} - 34 q^{64} - 32 q^{65} + 12 q^{66} - 24 q^{67} - 9 q^{68} - 18 q^{69} - 6 q^{70} + 42 q^{71} + 39 q^{72} - 8 q^{73} + 13 q^{74} - 6 q^{76} - 11 q^{77} + 24 q^{78} - 22 q^{79} + 52 q^{80} + 9 q^{81} - 96 q^{82} - 17 q^{83} + 18 q^{84} - 6 q^{85} + 21 q^{86} + 37 q^{88} - 72 q^{89} + 42 q^{90} - 24 q^{91} - 21 q^{92} + 9 q^{93} + 28 q^{94} - 4 q^{95} + 6 q^{97} + 72 q^{98} - 3 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 + 3 * q^3 - 6 * q^4 + 6 * q^5 - q^7 - 14 * q^8 - 3 * q^9 - 32 * q^10 + q^11 + 18 * q^12 + 8 * q^13 - q^14 + 12 * q^15 + 14 * q^16 + 12 * q^17 - 3 * q^18 - 2 * q^19 + 24 * q^20 + 11 * q^22 - 14 * q^23 - 39 * q^24 - 9 * q^25 - 16 * q^26 + 18 * q^28 - 9 * q^29 + 18 * q^30 - 3 * q^31 + 30 * q^32 - 6 * q^34 + 12 * q^35 - 36 * q^36 + 24 * q^37 - 4 * q^38 - 24 * q^39 + 12 * q^40 + 3 * q^41 - 12 * q^42 + 6 * q^43 - 48 * q^44 - 24 * q^45 + 18 * q^46 + 23 * q^47 - 6 * q^49 - 24 * q^50 + 27 * q^51 + 12 * q^52 + 28 * q^53 + 54 * q^54 - 32 * q^55 - 12 * q^56 + 3 * q^57 + 6 * q^58 + 3 * q^59 + 6 * q^62 + 3 * q^63 - 34 * q^64 - 32 * q^65 + 12 * q^66 - 24 * q^67 - 9 * q^68 - 18 * q^69 - 6 * q^70 + 42 * q^71 + 39 * q^72 - 8 * q^73 + 13 * q^74 - 6 * q^76 - 11 * q^77 + 24 * q^78 - 22 * q^79 + 52 * q^80 + 9 * q^81 - 96 * q^82 - 17 * q^83 + 18 * q^84 - 6 * q^85 + 21 * q^86 + 37 * q^88 - 72 * q^89 + 42 * q^90 - 24 * q^91 - 21 * q^92 + 9 * q^93 + 28 * q^94 - 4 * q^95 + 6 * q^97 + 72 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	99.2.m.a

Level	$99$
Weight	$2$
Character orbit	99.m
Analytic conductor	$0.791$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.790518980011\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}\)	\(-1 - \zeta_{15}^{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 97.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 4 T^{7} + 10 T^{6} + 26 T^{5} + \cdots + 1 \) T^8 + 4T^7 + 10T^6 + 26T^5 + 39T^4 - 14T^3 - 5T^2 - T + 1
$3$	\( T^{8} - 3 T^{7} + 6 T^{6} - 9 T^{5} + \cdots + 81 \) T^8 - 3T^7 + 6T^6 - 9T^5 + 9T^4 - 27T^3 + 54T^2 - 81*T + 81
$5$	\( T^{8} - 6 T^{7} + 20 T^{6} - 64 T^{5} + \cdots + 256 \) T^8 - 6T^7 + 20T^6 - 64T^5 + 144T^4 - 64T^3 - 256T + 256
$7$	\( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$11$	\( T^{8} - T^{7} - 20 T^{6} + T^{5} + \cdots + 14641 \) T^8 - T^7 - 20T^6 + T^5 + 309T^4 + 11T^3 - 2420T^2 - 1331*T + 14641
$13$	\( T^{8} - 8 T^{7} - 128 T^{5} + \cdots + 65536 \) T^8 - 8T^7 - 128T^5 + 2304T^4 - 8192T^3 + 20480T^2 - 49152T + 65536
$17$	\( (T^{4} - 6 T^{3} + 36 T^{2} - 81 T + 81)^{2} \) (T^4 - 6T^3 + 36T^2 - 81*T + 81)^2
$19$	\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$23$	\( (T^{4} + 7 T^{3} + 38 T^{2} + 77 T + 121)^{2} \) (T^4 + 7T^3 + 38T^2 + 77*T + 121)^2
$29$	\( T^{8} + 9 T^{7} + 45 T^{6} + \cdots + 6561 \) T^8 + 9T^7 + 45T^6 + 216T^5 + 729T^4 + 486T^3 + 4374T + 6561
$31$	\( T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1 \) T^8 + 3T^7 + 5T^6 + 8T^5 + 9T^4 + 2T^3 + 2T + 1
$37$	\( (T^{4} - 12 T^{3} + 94 T^{2} - 403 T + 961)^{2} \) (T^4 - 12T^3 + 94T^2 - 403*T + 961)^2
$41$	\( T^{8} - 3 T^{7} - 100 T^{6} + \cdots + 707281 \) T^8 - 3T^7 - 100T^6 - 1313T^5 + 12519T^4 + 58783T^3 + 151380T^2 + 414613*T + 707281
$43$	\( (T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81)^{2} \) (T^4 - 3T^3 + 18T^2 + 27*T + 81)^2
$47$	\( T^{8} - 23 T^{7} + 280 T^{6} + \cdots + 25411681 \) T^8 - 23T^7 + 280T^6 - 3313T^5 + 29199T^4 - 68657T^3 + 201640T^2 - 6084487*T + 25411681
$53$	\( (T^{4} - 14 T^{3} + 96 T^{2} - 319 T + 841)^{2} \) (T^4 - 14T^3 + 96T^2 - 319*T + 841)^2
$59$	\( T^{8} - 3 T^{7} + 5 T^{6} - 8 T^{5} + \cdots + 1 \) T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$61$	\( T^{8} - 90 T^{6} + 1350 T^{5} + \cdots + 4100625 \) T^8 - 90T^6 + 1350T^5 + 6075T^4 - 60750T^3 + 273375T^2 - 1366875T + 4100625
$67$	\( (T^{2} + 6 T + 36)^{4} \) (T^2 + 6*T + 36)^4
$71$	\( (T^{4} - 21 T^{3} + 166 T^{2} + 164 T + 1681)^{2} \) (T^4 - 21T^3 + 166T^2 + 164*T + 1681)^2
$73$	\( (T^{4} + 4 T^{3} + 46 T^{2} - 11 T + 1)^{2} \) (T^4 + 4T^3 + 46T^2 - 11*T + 1)^2
$79$	\( T^{8} + 22 T^{7} + \cdots + 104060401 \) T^8 + 22T^7 + 180T^6 + 2042T^5 + 31109T^4 + 257348T^3 + 2295225T^2 + 23696923*T + 104060401
$83$	\( T^{8} + 17 T^{7} + \cdots + 373301041 \) T^8 + 17T^7 + 130T^6 + 757T^5 - 10581T^4 - 502207T^3 - 2125310T^2 + 18799333*T + 373301041
$89$	\( (T^{2} + 18 T + 76)^{4} \) (T^2 + 18*T + 76)^4
$97$	\( T^{8} - 6 T^{7} + 216 T^{5} + \cdots + 1679616 \) T^8 - 6T^7 + 216T^5 - 1296T^4 + 7776T^3 - 279936*T + 1679616
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