LMFDB - Newform orbit 950.2.q.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(107,950)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([3, 10]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("950.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	950.q (of order $12$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.58578819202$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{24}$. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$\zeta_{24}^{6}$	$1 - \zeta_{24}^{4}$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

107.1

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

−0.965926

−

0.258819i

−0.517638

1.93185i

0.866025

0.500000i

1.00000

−

1.73205i

−1.22474

−

1.22474i

−0.707107

−

0.707107i

−0.866025

−

0.500000i

107.2

0.965926

0.258819i

0.517638

−

1.93185i

0.866025

0.500000i

1.00000

−

1.73205i

1.22474

1.22474i

0.707107

0.707107i

−0.866025

−

0.500000i

293.1

−0.965926

0.258819i

−0.517638

−

1.93185i

0.866025

−

0.500000i

1.00000

1.73205i

−1.22474

1.22474i

−0.707107

0.707107i

−0.866025

0.500000i

293.2

0.965926

−

0.258819i

0.517638

1.93185i

0.866025

−

0.500000i

1.00000

1.73205i

1.22474

−

1.22474i

0.707107

−

0.707107i

−0.866025

0.500000i

407.1

−0.258819

−

0.965926i

−1.93185

0.517638i

−0.866025

0.500000i

1.00000

1.73205i

−1.22474

−

1.22474i

0.707107

0.707107i

0.866025

−

0.500000i

407.2

0.258819

0.965926i

1.93185

−

0.517638i

−0.866025

0.500000i

1.00000

1.73205i

1.22474

1.22474i

−0.707107

−

0.707107i

0.866025

−

0.500000i

943.1

−0.258819

0.965926i

−1.93185

−

0.517638i

−0.866025

−

0.500000i

1.00000

−

1.73205i

−1.22474

1.22474i

0.707107

−

0.707107i

0.866025

0.500000i

943.2

0.258819

−

0.965926i

1.93185

0.517638i

−0.866025

−

0.500000i

1.00000

−

1.73205i

1.22474

−

1.22474i

−0.707107

0.707107i

0.866025

0.500000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
5.c	odd	4	2	inner
19.d	odd	6	1	inner
95.h	odd	6	1	inner
95.l	even	12	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	950.2.q.d	✓	8
5.b	even	2	1	inner	950.2.q.d	✓	8
5.c	odd	4	2	inner	950.2.q.d	✓	8
19.d	odd	6	1	inner	950.2.q.d	✓	8
95.h	odd	6	1	inner	950.2.q.d	✓	8
95.l	even	12	2	inner	950.2.q.d	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
950.2.q.d	✓	8	1.a	even	1	1	trivial
950.2.q.d	✓	8	5.b	even	2	1	inner
950.2.q.d	✓	8	5.c	odd	4	2	inner
950.2.q.d	✓	8	19.d	odd	6	1	inner
950.2.q.d	✓	8	95.h	odd	6	1	inner
950.2.q.d	✓	8	95.l	even	12	2	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(950, [\chi])$:

$ T_{3}^{8} - 16T_{3}^{4} + 256 $

$ T_{7}^{4} + 9 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$3$	$ T^{8} - 16T^{4} + 256 $ T^8 - 16*T^4 + 256
$5$	$ T^{8} $ T^8
$7$	$ (T^{4} + 9)^{2} $ (T^4 + 9)^2
$11$	$ (T - 3)^{8} $ (T - 3)^8
$13$	$ T^{8} - 16T^{4} + 256 $ T^8 - 16*T^4 + 256
$17$	$ T^{8} $ T^8
$19$	$ (T^{4} + 11 T^{2} + 361)^{2} $ (T^4 + 11*T^2 + 361)^2
$23$	$ T^{8} - 729 T^{4} + 531441 $ T^8 - 729*T^4 + 531441
$29$	$ T^{8} $ T^8
$31$	$ (T^{2} + 12)^{4} $ (T^2 + 12)^4
$37$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$41$	$ (T^{2} + 9 T + 27)^{4} $ (T^2 + 9*T + 27)^4
$43$	$ T^{8} - 2304 T^{4} + 5308416 $ T^8 - 2304*T^4 + 5308416
$47$	$ T^{8} - 11664 T^{4} + 136048896 $ T^8 - 11664*T^4 + 136048896
$53$	$ T^{8} - 81T^{4} + 6561 $ T^8 - 81*T^4 + 6561
$59$	$ (T^{4} + 108 T^{2} + 11664)^{2} $ (T^4 + 108*T^2 + 11664)^2
$61$	$ (T^{2} + 4 T + 16)^{4} $ (T^2 + 4*T + 16)^4
$67$	$ T^{8} - 16T^{4} + 256 $ T^8 - 16*T^4 + 256
$71$	$ (T^{2} + 18 T + 108)^{4} $ (T^2 + 18*T + 108)^4
$73$	$ T^{8} - 144 T^{4} + 20736 $ T^8 - 144*T^4 + 20736
$79$	$ (T^{4} + 12 T^{2} + 144)^{2} $ (T^4 + 12*T^2 + 144)^2
$83$	$ T^{8} $ T^8
$89$	$ (T^{4} + 27 T^{2} + 729)^{2} $ (T^4 + 27*T^2 + 729)^2
$97$	$ T^{8} - 4096 T^{4} + 16777216 $ T^8 - 4096*T^4 + 16777216
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.q (of order \(12\), degree \(4\), minimal)

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

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Newspace parameters

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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	950.2.q.d

Level	$950$
Weight	$2$
Character orbit	950.q
Analytic conductor	$7.586$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(\zeta_{24}^{6}\)	\(1 - \zeta_{24}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 107.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - T^{4} + 1 \) T^8 - T^4 + 1
$3$	\( T^{8} - 16T^{4} + 256 \) T^8 - 16*T^4 + 256
$5$	\( T^{8} \) T^8
$7$	\( (T^{4} + 9)^{2} \) (T^4 + 9)^2
$11$	\( (T - 3)^{8} \) (T - 3)^8
$13$	\( T^{8} - 16T^{4} + 256 \) T^8 - 16*T^4 + 256
$17$	\( T^{8} \) T^8
$19$	\( (T^{4} + 11 T^{2} + 361)^{2} \) (T^4 + 11*T^2 + 361)^2
$23$	\( T^{8} - 729 T^{4} + 531441 \) T^8 - 729*T^4 + 531441
$29$	\( T^{8} \) T^8
$31$	\( (T^{2} + 12)^{4} \) (T^2 + 12)^4
$37$	\( (T^{4} + 1)^{2} \) (T^4 + 1)^2
$41$	\( (T^{2} + 9 T + 27)^{4} \) (T^2 + 9*T + 27)^4
$43$	\( T^{8} - 2304 T^{4} + 5308416 \) T^8 - 2304*T^4 + 5308416
$47$	\( T^{8} - 11664 T^{4} + 136048896 \) T^8 - 11664*T^4 + 136048896
$53$	\( T^{8} - 81T^{4} + 6561 \) T^8 - 81*T^4 + 6561
$59$	\( (T^{4} + 108 T^{2} + 11664)^{2} \) (T^4 + 108*T^2 + 11664)^2
$61$	\( (T^{2} + 4 T + 16)^{4} \) (T^2 + 4*T + 16)^4
$67$	\( T^{8} - 16T^{4} + 256 \) T^8 - 16*T^4 + 256
$71$	\( (T^{2} + 18 T + 108)^{4} \) (T^2 + 18*T + 108)^4
$73$	\( T^{8} - 144 T^{4} + 20736 \) T^8 - 144*T^4 + 20736
$79$	\( (T^{4} + 12 T^{2} + 144)^{2} \) (T^4 + 12*T^2 + 144)^2
$83$	\( T^{8} \) T^8
$89$	\( (T^{4} + 27 T^{2} + 729)^{2} \) (T^4 + 27*T^2 + 729)^2
$97$	\( T^{8} - 4096 T^{4} + 16777216 \) T^8 - 4096*T^4 + 16777216
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