LMFDB - Newform orbit 625.2.e.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(124,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("625.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.e (of order $10$, degree $4$, not 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:	$4.99065012633$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-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 $\zeta_{20}$. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$\zeta_{20}^{6}$

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} $

124.1

−0.587785	+	0.809017i
0.587785	−	0.809017i
−0.951057	+	0.309017i
0.951057	−	0.309017i
−0.951057	−	0.309017i
0.951057	+	0.309017i
−0.587785	−	0.809017i
0.587785	+	0.809017i

−0.587785

−

0.190983i

−0.224514

0.309017i

−1.30902

−

0.951057i

0.190983

−

0.138757i

−

2.85410i

1.31433

1.80902i

0.881966

2.71441i

124.2

0.587785

0.190983i

0.224514

−

0.309017i

−1.30902

−

0.951057i

0.190983

−

0.138757i

2.85410i

−1.31433

−

1.80902i

0.881966

2.71441i

249.1

−0.951057

1.30902i

−2.48990

0.809017i

−0.190983

−

0.587785i

1.30902

−

4.02874i

−

3.85410i

−2.12663

−

0.690983i

3.11803

−

2.26538i

249.2

0.951057

−

1.30902i

2.48990

−

0.809017i

−0.190983

−

0.587785i

1.30902

−

4.02874i

3.85410i

2.12663

0.690983i

3.11803

−

2.26538i

374.1

−0.951057

−

1.30902i

−2.48990

−

0.809017i

−0.190983

0.587785i

1.30902

4.02874i

3.85410i

−2.12663

0.690983i

3.11803

2.26538i

374.2

0.951057

1.30902i

2.48990

0.809017i

−0.190983

0.587785i

1.30902

4.02874i

−

3.85410i

2.12663

−

0.690983i

3.11803

2.26538i

499.1

−0.587785

0.190983i

−0.224514

−

0.309017i

−1.30902

0.951057i

0.190983

0.138757i

2.85410i

1.31433

−

1.80902i

0.881966

−

2.71441i

499.2

0.587785

−

0.190983i

0.224514

0.309017i

−1.30902

0.951057i

0.190983

0.138757i

−

2.85410i

−1.31433

1.80902i

0.881966

−

2.71441i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
25.d	even	5	1	inner
25.e	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	625.2.e.e		8
5.b	even	2	1	inner	625.2.e.e		8
5.c	odd	4	1		625.2.d.c		4
5.c	odd	4	1		625.2.d.i		4
25.d	even	5	1		625.2.b.b		4
25.d	even	5	1	inner	625.2.e.e		8
25.d	even	5	2		625.2.e.f		8
25.e	even	10	1		625.2.b.b		4
25.e	even	10	1	inner	625.2.e.e		8
25.e	even	10	2		625.2.e.f		8
25.f	odd	20	1		625.2.a.a	✓	2
25.f	odd	20	1		625.2.a.d	yes	2
25.f	odd	20	1		625.2.d.c		4
25.f	odd	20	2		625.2.d.e		4
25.f	odd	20	2		625.2.d.f		4
25.f	odd	20	1		625.2.d.i		4
75.l	even	20	1		5625.2.a.c		2
75.l	even	20	1		5625.2.a.e		2
100.l	even	20	1		10000.2.a.b		2
100.l	even	20	1		10000.2.a.m		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
625.2.a.a	✓	2	25.f	odd	20	1
625.2.a.d	yes	2	25.f	odd	20	1
625.2.b.b		4	25.d	even	5	1
625.2.b.b		4	25.e	even	10	1
625.2.d.c		4	5.c	odd	4	1
625.2.d.c		4	25.f	odd	20	1
625.2.d.e		4	25.f	odd	20	2
625.2.d.f		4	25.f	odd	20	2
625.2.d.i		4	5.c	odd	4	1
625.2.d.i		4	25.f	odd	20	1
625.2.e.e		8	1.a	even	1	1	trivial
625.2.e.e		8	5.b	even	2	1	inner
625.2.e.e		8	25.d	even	5	1	inner
625.2.e.e		8	25.e	even	10	1	inner
625.2.e.f		8	25.d	even	5	2
625.2.e.f		8	25.e	even	10	2
5625.2.a.c		2	75.l	even	20	1
5625.2.a.e		2	75.l	even	20	1
10000.2.a.b		2	100.l	even	20	1
10000.2.a.m		2	100.l	even	20	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(625, [\chi])$:

$ T_{2}^{8} + T_{2}^{6} + 6T_{2}^{4} - 4T_{2}^{2} + 1 $

$ T_{3}^{8} - 11T_{3}^{6} + 46T_{3}^{4} + 4T_{3}^{2} + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + T^{6} + 6 T^{4} + \cdots + 1 $ T^8 + T^6 + 6T^4 - 4T^2 + 1
$3$	$ T^{8} - 11 T^{6} + \cdots + 1 $ T^8 - 11T^6 + 46T^4 + 4*T^2 + 1
$5$	$ T^{8} $ T^8
$7$	$ (T^{4} + 23 T^{2} + 121)^{2} $ (T^4 + 23*T^2 + 121)^2
$11$	$ (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 2T^3 + 4T^2 + 3*T + 1)^2
$13$	$ T^{8} - T^{6} + \cdots + 130321 $ T^8 - T^6 + 681T^4 - 14801T^2 + 130321
$17$	$ T^{8} + 31 T^{6} + \cdots + 14641 $ T^8 + 31T^6 + 3001T^4 - 10769*T^2 + 14641
$19$	$ (T^{4} - 10 T^{3} + \cdots + 25)^{2} $ (T^4 - 10T^3 + 40T^2 - 25*T + 25)^2
$23$	$ T^{8} - 36 T^{6} + \cdots + 6561 $ T^8 - 36T^6 + 486T^4 + 729*T^2 + 6561
$29$	$ (T^{4} + 10 T^{3} + \cdots + 400)^{2} $ (T^4 + 10T^3 + 40T^2 + 400)^2
$31$	$ (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} $ (T^4 + 2T^3 + 4T^2 + 8*T + 16)^2
$37$	$ T^{8} - 9 T^{6} + \cdots + 6561 $ T^8 - 9T^6 + 81T^4 - 729*T^2 + 6561
$41$	$ (T^{4} + 12 T^{3} + \cdots + 256)^{2} $ (T^4 + 12T^3 + 64T^2 + 128*T + 256)^2
$43$	$ (T^{4} + 72 T^{2} + 16)^{2} $ (T^4 + 72*T^2 + 16)^2
$47$	$ T^{8} - 164 T^{6} + \cdots + 141158161 $ T^8 - 164T^6 + 14086T^4 - 700979*T^2 + 141158161
$53$	$ T^{8} - 116 T^{6} + \cdots + 256 $ T^8 - 116T^6 + 5136T^4 + 704*T^2 + 256
$59$	$ (T^{4} + 25 T^{3} + \cdots + 3025)^{2} $ (T^4 + 25T^3 + 265T^2 + 825*T + 3025)^2
$61$	$ (T^{4} - 13 T^{3} + \cdots + 361)^{2} $ (T^4 - 13T^3 + 64T^2 + 38*T + 361)^2
$67$	$ T^{8} - 209 T^{6} + \cdots + 707281 $ T^8 - 209T^6 + 16521T^4 + 59711*T^2 + 707281
$71$	$ (T^{4} + 22 T^{3} + \cdots + 1681)^{2} $ (T^4 + 22T^3 + 204T^2 + 533*T + 1681)^2
$73$	$ T^{8} - 76 T^{6} + \cdots + 1 $ T^8 - 76T^6 + 2206T^4 + 29*T^2 + 1
$79$	$ (T^{4} - 15 T^{3} + \cdots + 2025)^{2} $ (T^4 - 15T^3 + 135T^2 - 675*T + 2025)^2
$83$	$ T^{8} - 256 T^{6} + \cdots + 16777216 $ T^8 - 256T^6 + 24576T^4 + 262144*T^2 + 16777216
$89$	$ (T^{4} - 5 T^{3} + 10 T^{2} + 25)^{2} $ (T^4 - 5T^3 + 10T^2 + 25)^2
$97$	$ T^{8} - 139 T^{6} + \cdots + 62742241 $ T^8 - 139T^6 + 9726T^4 - 348524*T^2 + 62742241
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.e (of order \(10\), degree \(4\), not minimal)

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

Introduction

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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	625.2.e.e

Level	$625$
Weight	$2$
Character orbit	625.e
Analytic conductor	$4.991$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.99065012633\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 124.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) T^8 + T^6 + 6T^4 - 4T^2 + 1
$3$	\( T^{8} - 11 T^{6} + \cdots + 1 \) T^8 - 11T^6 + 46T^4 + 4*T^2 + 1
$5$	\( T^{8} \) T^8
$7$	\( (T^{4} + 23 T^{2} + 121)^{2} \) (T^4 + 23*T^2 + 121)^2
$11$	\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 + 2T^3 + 4T^2 + 3*T + 1)^2
$13$	\( T^{8} - T^{6} + \cdots + 130321 \) T^8 - T^6 + 681T^4 - 14801T^2 + 130321
$17$	\( T^{8} + 31 T^{6} + \cdots + 14641 \) T^8 + 31T^6 + 3001T^4 - 10769*T^2 + 14641
$19$	\( (T^{4} - 10 T^{3} + \cdots + 25)^{2} \) (T^4 - 10T^3 + 40T^2 - 25*T + 25)^2
$23$	\( T^{8} - 36 T^{6} + \cdots + 6561 \) T^8 - 36T^6 + 486T^4 + 729*T^2 + 6561
$29$	\( (T^{4} + 10 T^{3} + \cdots + 400)^{2} \) (T^4 + 10T^3 + 40T^2 + 400)^2
$31$	\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) (T^4 + 2T^3 + 4T^2 + 8*T + 16)^2
$37$	\( T^{8} - 9 T^{6} + \cdots + 6561 \) T^8 - 9T^6 + 81T^4 - 729*T^2 + 6561
$41$	\( (T^{4} + 12 T^{3} + \cdots + 256)^{2} \) (T^4 + 12T^3 + 64T^2 + 128*T + 256)^2
$43$	\( (T^{4} + 72 T^{2} + 16)^{2} \) (T^4 + 72*T^2 + 16)^2
$47$	\( T^{8} - 164 T^{6} + \cdots + 141158161 \) T^8 - 164T^6 + 14086T^4 - 700979*T^2 + 141158161
$53$	\( T^{8} - 116 T^{6} + \cdots + 256 \) T^8 - 116T^6 + 5136T^4 + 704*T^2 + 256
$59$	\( (T^{4} + 25 T^{3} + \cdots + 3025)^{2} \) (T^4 + 25T^3 + 265T^2 + 825*T + 3025)^2
$61$	\( (T^{4} - 13 T^{3} + \cdots + 361)^{2} \) (T^4 - 13T^3 + 64T^2 + 38*T + 361)^2
$67$	\( T^{8} - 209 T^{6} + \cdots + 707281 \) T^8 - 209T^6 + 16521T^4 + 59711*T^2 + 707281
$71$	\( (T^{4} + 22 T^{3} + \cdots + 1681)^{2} \) (T^4 + 22T^3 + 204T^2 + 533*T + 1681)^2
$73$	\( T^{8} - 76 T^{6} + \cdots + 1 \) T^8 - 76T^6 + 2206T^4 + 29*T^2 + 1
$79$	\( (T^{4} - 15 T^{3} + \cdots + 2025)^{2} \) (T^4 - 15T^3 + 135T^2 - 675*T + 2025)^2
$83$	\( T^{8} - 256 T^{6} + \cdots + 16777216 \) T^8 - 256T^6 + 24576T^4 + 262144*T^2 + 16777216
$89$	\( (T^{4} - 5 T^{3} + 10 T^{2} + 25)^{2} \) (T^4 - 5T^3 + 10T^2 + 25)^2
$97$	\( T^{8} - 139 T^{6} + \cdots + 62742241 \) T^8 - 139T^6 + 9726T^4 - 348524*T^2 + 62742241
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\(n\)	\(2\)
\(\chi(n)\)	\(\zeta_{20}^{6}\)