LMFDB - Newform orbit 1950.2.y.l

Newspace parameters

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.y (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$1 - \zeta_{24}^{4}$	$1$	$-1$

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} $

49.1

−0.258819	+	0.965926i
0.258819	−	0.965926i
−0.965926	−	0.258819i
0.965926	+	0.258819i
−0.258819	−	0.965926i
0.258819	+	0.965926i
−0.965926	+	0.258819i
0.965926	−	0.258819i

0.500000

0.866025i

−0.866025

0.500000i

−0.500000

0.866025i

−0.866025

−

0.500000i

−0.465926

0.807007i

−1.00000

0.500000

−

0.866025i

49.2

0.500000

0.866025i

−0.866025

0.500000i

−0.500000

0.866025i

−0.866025

−

0.500000i

1.46593

−

2.53906i

−1.00000

0.500000

−

0.866025i

49.3

0.500000

0.866025i

0.866025

−

0.500000i

−0.500000

0.866025i

0.866025

0.500000i

0.241181

−

0.417738i

−1.00000

0.500000

−

0.866025i

49.4

0.500000

0.866025i

0.866025

−

0.500000i

−0.500000

0.866025i

0.866025

0.500000i

0.758819

−

1.31431i

−1.00000

0.500000

−

0.866025i

199.1

0.500000

−

0.866025i

−0.866025

−

0.500000i

−0.500000

−

0.866025i

−0.866025

0.500000i

−0.465926

−

0.807007i

−1.00000

0.500000

0.866025i

199.2

0.500000

−

0.866025i

−0.866025

−

0.500000i

−0.500000

−

0.866025i

−0.866025

0.500000i

1.46593

2.53906i

−1.00000

0.500000

0.866025i

199.3

0.500000

−

0.866025i

0.866025

0.500000i

−0.500000

−

0.866025i

0.866025

−

0.500000i

0.241181

0.417738i

−1.00000

0.500000

0.866025i

199.4

0.500000

−

0.866025i

0.866025

0.500000i

−0.500000

−

0.866025i

0.866025

−

0.500000i

0.758819

1.31431i

−1.00000

0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1950.2.y.l		8
5.b	even	2	1		1950.2.y.i		8
5.c	odd	4	1		1950.2.bc.e	✓	8
5.c	odd	4	1		1950.2.bc.f	yes	8
13.e	even	6	1		1950.2.y.i		8
65.l	even	6	1	inner	1950.2.y.l		8
65.r	odd	12	1		1950.2.bc.e	✓	8
65.r	odd	12	1		1950.2.bc.f	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1950.2.y.i		8	5.b	even	2	1
1950.2.y.i		8	13.e	even	6	1
1950.2.y.l		8	1.a	even	1	1	trivial
1950.2.y.l		8	65.l	even	6	1	inner
1950.2.bc.e	✓	8	5.c	odd	4	1
1950.2.bc.e	✓	8	65.r	odd	12	1
1950.2.bc.f	yes	8	5.c	odd	4	1
1950.2.bc.f	yes	8	65.r	odd	12	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} - 4T_{7}^{7} + 14T_{7}^{6} - 16T_{7}^{5} + 22T_{7}^{4} - 8T_{7}^{3} + 20T_{7}^{2} - 8T_{7} + 4 $ T7^8 - 4*T7^7 + 14*T7^6 - 16*T7^5 + 22*T7^4 - 8*T7^3 + 20*T7^2 - 8*T7 + 4 acting on $S_{2}^{\mathrm{new}}(1950, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - T + 1)^{4} $ (T^2 - T + 1)^4
$3$	$ (T^{4} - T^{2} + 1)^{2} $ (T^4 - T^2 + 1)^2
$5$	$ T^{8} $ T^8
$7$	$ T^{8} - 4 T^{7} + 14 T^{6} - 16 T^{5} + \cdots + 4 $ T^8 - 4T^7 + 14T^6 - 16T^5 + 22T^4 - 8T^3 + 20T^2 - 8*T + 4
$11$	$ T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 529 $ T^8 + 12T^7 + 54T^6 + 72T^5 - 85T^4 - 216T^3 + 294T^2 + 828*T + 529
$13$	$ T^{8} + 191 T^{4} + 28561 $ T^8 + 191*T^4 + 28561
$17$	$ T^{8} - 38 T^{6} + 1398 T^{4} + \cdots + 2116 $ T^8 - 38T^6 + 1398T^4 - 4104T^3 + 2140T^2 + 4968*T + 2116
$19$	$ T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 324 $ T^8 + 12T^7 + 54T^6 + 72T^5 - 90T^4 - 216T^3 + 324T^2 + 648*T + 324
$23$	$ T^{8} - 12 T^{7} + 12 T^{6} + \cdots + 58081 $ T^8 - 12T^7 + 12T^6 + 432T^5 - 433T^4 - 13392T^3 + 37452T^2 + 89652*T + 58081
$29$	$ T^{8} + 4 T^{7} + 100 T^{6} + \cdots + 341056 $ T^8 + 4T^7 + 100T^6 + 592T^5 + 9496T^4 + 43648T^3 + 166240T^2 + 270976*T + 341056
$31$	$ T^{8} + 204 T^{6} + 12128 T^{4} + \cdots + 145924 $ T^8 + 204T^6 + 12128T^4 + 171672*T^2 + 145924
$37$	$ T^{8} - 16 T^{7} + 260 T^{6} + \cdots + 16056049 $ T^8 - 16T^7 + 260T^6 - 2032T^5 + 20791T^4 - 132416T^3 + 1082276T^2 - 4199336*T + 16056049
$41$	$ T^{8} - 102 T^{6} + 10550 T^{4} + \cdots + 21316 $ T^8 - 102T^6 + 10550T^4 - 42840T^3 + 73692T^2 - 61320*T + 21316
$43$	$ T^{8} + 24 T^{7} + 186 T^{6} + \cdots + 386884 $ T^8 + 24T^7 + 186T^6 - 144T^5 - 5290T^4 + 3528T^3 + 111516T^2 - 365736*T + 386884
$47$	$ (T^{4} + 16 T^{3} + 20 T^{2} - 544 T - 1724)^{2} $ (T^4 + 16T^3 + 20T^2 - 544*T - 1724)^2
$53$	$ T^{8} + 76 T^{6} + 1824 T^{4} + \cdots + 9604 $ T^8 + 76T^6 + 1824T^4 + 13720*T^2 + 9604
$59$	$ T^{8} - 12 T^{7} - 64 T^{6} + \cdots + 5740816 $ T^8 - 12T^7 - 64T^6 + 1344T^5 + 7356T^4 - 212352T^3 + 1466624T^2 - 4542816*T + 5740816
$61$	$ T^{8} + 16 T^{7} + 292 T^{6} + \cdots + 8567329 $ T^8 + 16T^7 + 292T^6 + 1888T^5 + 23935T^4 + 138016T^3 + 1412452T^2 + 3606064*T + 8567329
$67$	$ T^{8} + 24 T^{7} + 448 T^{6} + \cdots + 20647936 $ T^8 + 24T^7 + 448T^6 + 4224T^5 + 34752T^4 + 144384T^3 + 913408T^2 + 2617344*T + 20647936
$71$	$ T^{8} - 60 T^{7} + 1596 T^{6} + \cdots + 6115729 $ T^8 - 60T^7 + 1596T^6 - 23760T^5 + 213623T^4 - 1173744T^3 + 3907740T^2 - 7329972*T + 6115729
$73$	$ (T^{4} - 12 T^{3} - 34 T^{2} + 804 T - 2231)^{2} $ (T^4 - 12T^3 - 34T^2 + 804*T - 2231)^2
$79$	$ (T^{4} - 4 T^{3} - 102 T^{2} + 212 T + 622)^{2} $ (T^4 - 4T^3 - 102T^2 + 212*T + 622)^2
$83$	$ (T^{4} - 4 T^{3} - 70 T^{2} + 340 T - 263)^{2} $ (T^4 - 4T^3 - 70T^2 + 340*T - 263)^2
$89$	$ T^{8} + 24 T^{7} + 138 T^{6} + \cdots + 5080516 $ T^8 + 24T^7 + 138T^6 - 1296T^5 - 8074T^4 + 58968T^3 + 275772T^2 - 2461368*T + 5080516
$97$	$ T^{8} + 214 T^{6} + \cdots + 61606801 $ T^8 + 214T^6 - 672T^5 + 37947T^4 - 71904T^3 + 1792582T^2 + 2637264T + 61606801
show more
show less

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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.y (of order \(6\), degree \(2\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1950.2.y.l

Level	$1950$
Weight	$2$
Character orbit	1950.y
Analytic conductor	$15.571$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(15.5708283941\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(n\)	\(301\)	\(1301\)	\(1327\)
\(\chi(n)\)	\(1 - \zeta_{24}^{4}\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 199.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{4} \) (T^2 - T + 1)^4
$3$	\( (T^{4} - T^{2} + 1)^{2} \) (T^4 - T^2 + 1)^2
$5$	\( T^{8} \) T^8
$7$	\( T^{8} - 4 T^{7} + 14 T^{6} - 16 T^{5} + \cdots + 4 \) T^8 - 4T^7 + 14T^6 - 16T^5 + 22T^4 - 8T^3 + 20T^2 - 8*T + 4
$11$	\( T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 529 \) T^8 + 12T^7 + 54T^6 + 72T^5 - 85T^4 - 216T^3 + 294T^2 + 828*T + 529
$13$	\( T^{8} + 191 T^{4} + 28561 \) T^8 + 191*T^4 + 28561
$17$	\( T^{8} - 38 T^{6} + 1398 T^{4} + \cdots + 2116 \) T^8 - 38T^6 + 1398T^4 - 4104T^3 + 2140T^2 + 4968*T + 2116
$19$	\( T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 324 \) T^8 + 12T^7 + 54T^6 + 72T^5 - 90T^4 - 216T^3 + 324T^2 + 648*T + 324
$23$	\( T^{8} - 12 T^{7} + 12 T^{6} + \cdots + 58081 \) T^8 - 12T^7 + 12T^6 + 432T^5 - 433T^4 - 13392T^3 + 37452T^2 + 89652*T + 58081
$29$	\( T^{8} + 4 T^{7} + 100 T^{6} + \cdots + 341056 \) T^8 + 4T^7 + 100T^6 + 592T^5 + 9496T^4 + 43648T^3 + 166240T^2 + 270976*T + 341056
$31$	\( T^{8} + 204 T^{6} + 12128 T^{4} + \cdots + 145924 \) T^8 + 204T^6 + 12128T^4 + 171672*T^2 + 145924
$37$	\( T^{8} - 16 T^{7} + 260 T^{6} + \cdots + 16056049 \) T^8 - 16T^7 + 260T^6 - 2032T^5 + 20791T^4 - 132416T^3 + 1082276T^2 - 4199336*T + 16056049
$41$	\( T^{8} - 102 T^{6} + 10550 T^{4} + \cdots + 21316 \) T^8 - 102T^6 + 10550T^4 - 42840T^3 + 73692T^2 - 61320*T + 21316
$43$	\( T^{8} + 24 T^{7} + 186 T^{6} + \cdots + 386884 \) T^8 + 24T^7 + 186T^6 - 144T^5 - 5290T^4 + 3528T^3 + 111516T^2 - 365736*T + 386884
$47$	\( (T^{4} + 16 T^{3} + 20 T^{2} - 544 T - 1724)^{2} \) (T^4 + 16T^3 + 20T^2 - 544*T - 1724)^2
$53$	\( T^{8} + 76 T^{6} + 1824 T^{4} + \cdots + 9604 \) T^8 + 76T^6 + 1824T^4 + 13720*T^2 + 9604
$59$	\( T^{8} - 12 T^{7} - 64 T^{6} + \cdots + 5740816 \) T^8 - 12T^7 - 64T^6 + 1344T^5 + 7356T^4 - 212352T^3 + 1466624T^2 - 4542816*T + 5740816
$61$	\( T^{8} + 16 T^{7} + 292 T^{6} + \cdots + 8567329 \) T^8 + 16T^7 + 292T^6 + 1888T^5 + 23935T^4 + 138016T^3 + 1412452T^2 + 3606064*T + 8567329
$67$	\( T^{8} + 24 T^{7} + 448 T^{6} + \cdots + 20647936 \) T^8 + 24T^7 + 448T^6 + 4224T^5 + 34752T^4 + 144384T^3 + 913408T^2 + 2617344*T + 20647936
$71$	\( T^{8} - 60 T^{7} + 1596 T^{6} + \cdots + 6115729 \) T^8 - 60T^7 + 1596T^6 - 23760T^5 + 213623T^4 - 1173744T^3 + 3907740T^2 - 7329972*T + 6115729
$73$	\( (T^{4} - 12 T^{3} - 34 T^{2} + 804 T - 2231)^{2} \) (T^4 - 12T^3 - 34T^2 + 804*T - 2231)^2
$79$	\( (T^{4} - 4 T^{3} - 102 T^{2} + 212 T + 622)^{2} \) (T^4 - 4T^3 - 102T^2 + 212*T + 622)^2
$83$	\( (T^{4} - 4 T^{3} - 70 T^{2} + 340 T - 263)^{2} \) (T^4 - 4T^3 - 70T^2 + 340*T - 263)^2
$89$	\( T^{8} + 24 T^{7} + 138 T^{6} + \cdots + 5080516 \) T^8 + 24T^7 + 138T^6 - 1296T^5 - 8074T^4 + 58968T^3 + 275772T^2 - 2461368*T + 5080516
$97$	\( T^{8} + 214 T^{6} + \cdots + 61606801 \) T^8 + 214T^6 - 672T^5 + 37947T^4 - 71904T^3 + 1792582T^2 + 2637264T + 61606801
show more
show less