LMFDB - Newform orbit 1120.2.g.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(449,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1120.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.g (of order $2$, degree $1$, 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:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.65174749855744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 $ x^10 + 13x^8 + 56x^6 + 97x^4 + 61x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 basis $1,\beta_1,\ldots,\beta_{9}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{5} q^{7} + (\beta_{9} - \beta_{8} - \beta_{3} - 1) q^{9} + (\beta_{9} - \beta_{8} + \beta_{2} - 1) q^{11} + (\beta_{7} + \beta_{5} - 2 \beta_1) q^{13} + ( - \beta_{9} + \beta_{7} + \beta_{6} + \cdots - 1) q^{15}+ \cdots + (4 \beta_{3} + 2 \beta_{2} + 6) q^{99}+O(q^{100}) $ q + b1 * q^3 + b4 * q^5 + b5 * q^7 + (b9 - b8 - b3 - 1) * q^9 + (b9 - b8 + b2 - 1) * q^11 + (b7 + b5 - 2b1) q^13 + (-b9 + b7 + b6 + b3 - b1 - 1) * q^15 + (-b9 - b8 + b7 - b5) * q^17 + (b6 - b4 + 2) * q^19 - b3 * q^21 + (-b9 - b8 + b6 - 2b5 + b4) q^23 + (-b9 + b7 - b4 - b3 - b2 + b1 + 1) * q^25 + (b9 + b8 + 2b6 - 2b5 + 2b4 - b1) q^27 + (-b6 + b4 + b3 + 2b2 + 2) q^29 + (2b6 - 2b4 - 2b3 - 2b2 - 2) * q^31 + (-b7 + b6 - b5 + b4 - 2b1) q^33 + b8 * q^35 + (b9 + b8 + b6 - 2b5 + b4) q^37 + (-2b9 + 2b8 + b6 - b4 + 2b3 + b2 + 5) q^39 - 2b2 q^41 + (2b9 + 2b8 - 2b7 + 2b6 + 2b5 + 2b4) * q^43 + (-b9 + b8 - b7 + 3b5 - b4 + 2b3 + 2b1) q^45 + (-b9 - b8 + 2b5 + b1) q^47 - q^49 + (b9 - b8 + 2b6 - 2b4 - b2 - 3) * q^51 + (-b9 - b8 - b6 + 2b5 - b4 - 2b1) * q^53 + (-b9 + b8 - 2b7 + 2b6 + 2b5 + b1 - 2) q^55 + (b9 + b8 - 2b7 - b6 - b4 + 4b1) * q^57 + (2b9 - 2b8 + b6 - b4 - 2b3 - 2b2 + 4) * q^59 + (-b6 + b4 + 2b2 - 2) q^61 + (b6 - b5 + b4 - b1) * q^63 + (3b9 - b7 - b6 + 2b5 - 2b3 + b2 + 3b1) * q^65 + (2b6 + 2b4 - 2b1) q^67 + (2b6 - 2b4 + 2b3 - 2b2 - 2) * q^69 + (-b9 + b8 - b6 + b4 - 2b3 + 2b2) * q^71 + (-b9 - b8 - b6 - 2b5 - b4 - 2b1) * q^73 + (3b9 - b8 + b6 - 2b5 - b4 - 2b3 + b1 - 6) q^75 + (-b7 + b6 - b5 + b4) * q^77 + (b6 - b4 - 2b3 - b2 + 7) q^79 + (-b9 + b8 + b6 - b4 + 6b3 + 2b2 - 3) * q^81 + (-b9 - b8 + 4b7 - 2b6 + 6b5 - 2b4 - 2b1) q^83 + (b9 - b8 + 2b6 - 4b5 + b3 + 2b1 - 2) q^85 + (-3b9 - 3b8 - 2b5 + 3b1) * q^87 + (-2b9 + 2b8 + 2b6 - 2b4 - 2b3 - 2b2) * q^89 + (2b3 + b2 - 1) q^91 + (4b9 + 4b8 - 2b7 - 2b5 - 2b1) q^93 + (b9 - b7 + 3b4 + b3 + b2 - b1 - 6) q^95 + (b9 + b8 + b7 - 2b6 + 3b5 - 2b4) q^97 + (4b3 + 2b2 + 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65}+ \cdots + 80 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 $ x^10 + 13*x^8 + 56*x^6 + 97*x^4 + 61*x^2 + 4 :

$\beta_{1}$	$=$	$ ( \nu^{9} + 12\nu^{7} + 44\nu^{5} + 55\nu^{3} + 20\nu ) / 2 $ (v^9 + 12v^7 + 44v^5 + 55v^3 + 20v) / 2
$\beta_{2}$	$=$	$ ( -3\nu^{8} - 34\nu^{6} - 112\nu^{4} - 107\nu^{2} - 2 ) / 2 $ (-3v^8 - 34v^6 - 112v^4 - 107v^2 - 2) / 2
$\beta_{3}$	$=$	$ ( -3\nu^{8} - 34\nu^{6} - 112\nu^{4} - 111\nu^{2} - 12 ) / 2 $ (-3v^8 - 34v^6 - 112v^4 - 111v^2 - 12) / 2
$\beta_{4}$	$=$	$ ( -3\nu^{9} - 3\nu^{8} - 34\nu^{7} - 34\nu^{6} - 110\nu^{5} - 110\nu^{4} - 93\nu^{3} - 97\nu^{2} + 16\nu ) / 4 $ (-3v^9 - 3v^8 - 34v^7 - 34v^6 - 110v^5 - 110v^4 - 93v^3 - 97v^2 + 16*v) / 4
$\beta_{5}$	$=$	$ ( -2\nu^{9} - 23\nu^{7} - 78\nu^{5} - 82\nu^{3} - 13\nu ) / 2 $ (-2v^9 - 23v^7 - 78v^5 - 82v^3 - 13*v) / 2
$\beta_{6}$	$=$	$ ( -3\nu^{9} + 3\nu^{8} - 34\nu^{7} + 34\nu^{6} - 110\nu^{5} + 110\nu^{4} - 93\nu^{3} + 97\nu^{2} + 16\nu ) / 4 $ (-3v^9 + 3v^8 - 34v^7 + 34v^6 - 110v^5 + 110v^4 - 93v^3 + 97v^2 + 16*v) / 4
$\beta_{7}$	$=$	$ ( 3\nu^{9} + 35\nu^{7} + 122\nu^{5} + 137\nu^{3} + 29\nu ) / 2 $ (3v^9 + 35v^7 + 122v^5 + 137v^3 + 29*v) / 2
$\beta_{8}$	$=$	$ ( 5\nu^{9} + 5\nu^{8} + 56\nu^{7} + 58\nu^{6} + 178\nu^{5} + 198\nu^{4} + 151\nu^{3} + 203\nu^{2} - 14\nu + 20 ) / 4 $ (5v^9 + 5v^8 + 56v^7 + 58v^6 + 178v^5 + 198v^4 + 151v^3 + 203v^2 - 14*v + 20) / 4
$\beta_{9}$	$=$	$ ( 5\nu^{9} - 5\nu^{8} + 56\nu^{7} - 58\nu^{6} + 178\nu^{5} - 198\nu^{4} + 151\nu^{3} - 203\nu^{2} - 14\nu - 20 ) / 4 $ (5v^9 - 5v^8 + 56v^7 - 58v^6 + 178v^5 - 198v^4 + 151v^3 - 203v^2 - 14*v - 20) / 4

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -\beta_{7} - \beta_{5} + \beta_1 ) / 2 $ (-b7 - b5 + b1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{3} + \beta_{2} - 5 ) / 2 $ (-b3 + b2 - 5) / 2
$\nu^{3}$	$=$	$ ( \beta_{9} + \beta_{8} + 4\beta_{7} + \beta_{6} + 6\beta_{5} + \beta_{4} - 2\beta_1 ) / 2 $ (b9 + b8 + 4b7 + b6 + 6b5 + b4 - 2*b1) / 2
$\nu^{4}$	$=$	$ ( -2\beta_{6} + 2\beta_{4} + 5\beta_{3} - 7\beta_{2} + 23 ) / 2 $ (-2b6 + 2b4 + 5b3 - 7b2 + 23) / 2
$\nu^{5}$	$=$	$ ( -8\beta_{9} - 8\beta_{8} - 21\beta_{7} - 6\beta_{6} - 41\beta_{5} - 6\beta_{4} + 3\beta_1 ) / 2 $ (-8b9 - 8b8 - 21b7 - 6b6 - 41b5 - 6b4 + 3*b1) / 2
$\nu^{6}$	$=$	$ ( -3\beta_{9} + 3\beta_{8} + 17\beta_{6} - 17\beta_{4} - 24\beta_{3} + 46\beta_{2} - 128 ) / 2 $ (-3b9 + 3b8 + 17b6 - 17b4 - 24b3 + 46b2 - 128) / 2
$\nu^{7}$	$=$	$ ( 52\beta_{9} + 52\beta_{8} + 125\beta_{7} + 32\beta_{6} + 273\beta_{5} + 32\beta_{4} + 7\beta_1 ) / 2 $ (52b9 + 52b8 + 125b7 + 32b6 + 273b5 + 32b4 + 7*b1) / 2
$\nu^{8}$	$=$	$ ( 34\beta_{9} - 34\beta_{8} - 118\beta_{6} + 118\beta_{4} + 121\beta_{3} - 297\beta_{2} + 769 ) / 2 $ (34b9 - 34b8 - 118b6 + 118b4 + 121b3 - 297b2 + 769) / 2
$\nu^{9}$	$=$	$ ( -327\beta_{9} - 327\beta_{8} - 776\beta_{7} - 175\beta_{6} - 1782\beta_{5} - 175\beta_{4} - 122\beta_1 ) / 2 $ (-327b9 - 327b8 - 776b7 - 175b6 - 1782b5 - 175b4 - 122*b1) / 2

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$1$	$1$	$-1$

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.

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} $

449.1

	−	1.84576i
	−	0.271831i
		2.52064i
	−	1.28447i
	−	1.23118i
		1.23118i
		1.28447i
	−	2.52064i
		0.271831i
		1.84576i

−

3.25260i

−1.49436

1.66340i

−

1.00000i

−7.57939

449.2

−

2.19794i

1.64514

−

1.51444i

1.00000i

−1.83094

449.3

−

1.83297i

1.86302

−

1.23660i

−

1.00000i

−0.359777

449.4

−

1.63460i

−2.23081

−

0.153266i

1.00000i

0.328072

449.5

−

0.746976i

−0.782984

−

2.09450i

−

1.00000i

2.44203

449.6

0.746976i

−0.782984

2.09450i

1.00000i

2.44203

449.7

1.63460i

−2.23081

0.153266i

−

1.00000i

0.328072

449.8

1.83297i

1.86302

1.23660i

1.00000i

−0.359777

449.9

2.19794i

1.64514

1.51444i

−

1.00000i

−1.83094

449.10

3.25260i

−1.49436

−

1.66340i

1.00000i

−7.57939

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1120.2.g.b	✓	10
4.b	odd	2	1		1120.2.g.c	yes	10
5.b	even	2	1	inner	1120.2.g.b	✓	10
5.c	odd	4	1		5600.2.a.bu		5
5.c	odd	4	1		5600.2.a.bw		5
8.b	even	2	1		2240.2.g.o		10
8.d	odd	2	1		2240.2.g.n		10
20.d	odd	2	1		1120.2.g.c	yes	10
20.e	even	4	1		5600.2.a.bv		5
20.e	even	4	1		5600.2.a.bx		5
40.e	odd	2	1		2240.2.g.n		10
40.f	even	2	1		2240.2.g.o		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1120.2.g.b	✓	10	1.a	even	1	1	trivial
1120.2.g.b	✓	10	5.b	even	2	1	inner
1120.2.g.c	yes	10	4.b	odd	2	1
1120.2.g.c	yes	10	20.d	odd	2	1
2240.2.g.n		10	8.d	odd	2	1
2240.2.g.n		10	40.e	odd	2	1
2240.2.g.o		10	8.b	even	2	1
2240.2.g.o		10	40.f	even	2	1
5600.2.a.bu		5	5.c	odd	4	1
5600.2.a.bv		5	20.e	even	4	1
5600.2.a.bw		5	5.c	odd	4	1
5600.2.a.bx		5	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1120, [\chi])$:

$ T_{3}^{10} + 22T_{3}^{8} + 165T_{3}^{6} + 532T_{3}^{4} + 708T_{3}^{2} + 256 $

$ T_{11}^{5} + 4T_{11}^{4} - 25T_{11}^{3} - 152T_{11}^{2} - 248T_{11} - 128 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} $ T^10
$3$	$ T^{10} + 22 T^{8} + \cdots + 256 $ T^10 + 22T^8 + 165T^6 + 532T^4 + 708T^2 + 256
$5$	$ T^{10} + 2 T^{9} + \cdots + 3125 $ T^10 + 2T^9 - T^8 - 4T^7 + 20T^6 + 116T^5 + 100T^4 - 100T^3 - 125T^2 + 1250T + 3125
$7$	$ (T^{2} + 1)^{5} $ (T^2 + 1)^5
$11$	$ (T^{5} + 4 T^{4} + \cdots - 128)^{2} $ (T^5 + 4T^4 - 25T^3 - 152T^2 - 248T - 128)^2
$13$	$ T^{10} + 94 T^{8} + \cdots + 678976 $ T^10 + 94T^8 + 2965T^6 + 41932T^4 + 274724T^2 + 678976
$17$	$ T^{10} + 66 T^{8} + \cdots + 1024 $ T^10 + 66T^8 + 937T^6 + 3944T^4 + 4496T^2 + 1024
$19$	$ (T^{5} - 12 T^{4} + \cdots - 256)^{2} $ (T^5 - 12T^4 + 30T^3 + 72T^2 - 184T - 256)^2
$23$	$ T^{10} + 104 T^{8} + \cdots + 262144 $ T^10 + 104T^8 + 2832T^6 + 31488T^4 + 151552T^2 + 262144
$29$	$ (T^{5} - 12 T^{4} + \cdots - 4744)^{2} $ (T^5 - 12T^4 - 15T^3 + 462T^2 - 124T - 4744)^2
$31$	$ (T^{5} + 12 T^{4} + \cdots - 4096)^{2} $ (T^5 + 12T^4 - 56T^3 - 1184T^2 - 4480T - 4096)^2
$37$	$ T^{10} + 136 T^{8} + \cdots + 65536 $ T^10 + 136T^8 + 5648T^6 + 73728T^4 + 196608T^2 + 65536
$41$	$ (T^{5} + 2 T^{4} + \cdots - 128)^{2} $ (T^5 + 2T^4 - 100T^3 - 632T^2 - 1056T - 128)^2
$43$	$ T^{10} + \cdots + 228130816 $ T^10 + 344T^8 + 41040T^6 + 2136320T^4 + 45581312T^2 + 228130816
$47$	$ T^{10} + 122 T^{8} + \cdots + 4096 $ T^10 + 122T^8 + 4617T^6 + 69392T^4 + 358976T^2 + 4096
$53$	$ T^{10} + 208 T^{8} + \cdots + 3444736 $ T^10 + 208T^8 + 13920T^6 + 333312T^4 + 2001152T^2 + 3444736
$59$	$ (T^{5} - 16 T^{4} + \cdots - 58112)^{2} $ (T^5 - 16T^4 - 106T^3 + 2048T^2 + 2232T - 58112)^2
$61$	$ (T^{5} + 10 T^{4} + \cdots + 1648)^{2} $ (T^5 + 10T^4 - 46T^3 - 468T^2 + 72T + 1648)^2
$67$	$ T^{10} + 232 T^{8} + \cdots + 262144 $ T^10 + 232T^8 + 10768T^6 + 101888T^4 + 299008T^2 + 262144
$71$	$ (T^{5} + 4 T^{4} + \cdots - 1024)^{2} $ (T^5 + 4T^4 - 228T^3 - 1344T^2 - 2240T - 1024)^2
$73$	$ T^{10} + 224 T^{8} + \cdots + 4194304 $ T^10 + 224T^8 + 13792T^6 + 306688T^4 + 2052352T^2 + 4194304
$79$	$ (T^{5} - 32 T^{4} + \cdots - 3904)^{2} $ (T^5 - 32T^4 + 363T^3 - 1884T^2 + 4500T - 3904)^2
$83$	$ T^{10} + \cdots + 750321664 $ T^10 + 684T^8 + 155812T^6 + 12747648T^4 + 192013312T^2 + 750321664
$89$	$ (T^{5} + 2 T^{4} + \cdots + 3104)^{2} $ (T^5 + 2T^4 - 168T^3 - 400T^2 + 1936T + 3104)^2
$97$	$ T^{10} + 274 T^{8} + \cdots + 11723776 $ T^10 + 274T^8 + 26681T^6 + 1074792T^4 + 14907024T^2 + 11723776
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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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{5} q^{7} + (\beta_{9} - \beta_{8} - \beta_{3} - 1) q^{9} + (\beta_{9} - \beta_{8} + \beta_{2} - 1) q^{11} + (\beta_{7} + \beta_{5} - 2 \beta_1) q^{13} + ( - \beta_{9} + \beta_{7} + \beta_{6} + \cdots - 1) q^{15}+ \cdots + (4 \beta_{3} + 2 \beta_{2} + 6) q^{99}+O(q^{100}) \) q + b1 * q^3 + b4 * q^5 + b5 * q^7 + (b9 - b8 - b3 - 1) * q^9 + (b9 - b8 + b2 - 1) * q^11 + (b7 + b5 - 2b1) q^13 + (-b9 + b7 + b6 + b3 - b1 - 1) * q^15 + (-b9 - b8 + b7 - b5) * q^17 + (b6 - b4 + 2) * q^19 - b3 * q^21 + (-b9 - b8 + b6 - 2b5 + b4) q^23 + (-b9 + b7 - b4 - b3 - b2 + b1 + 1) * q^25 + (b9 + b8 + 2b6 - 2b5 + 2b4 - b1) q^27 + (-b6 + b4 + b3 + 2b2 + 2) q^29 + (2b6 - 2b4 - 2b3 - 2b2 - 2) * q^31 + (-b7 + b6 - b5 + b4 - 2b1) q^33 + b8 * q^35 + (b9 + b8 + b6 - 2b5 + b4) q^37 + (-2b9 + 2b8 + b6 - b4 + 2b3 + b2 + 5) q^39 - 2b2 q^41 + (2b9 + 2b8 - 2b7 + 2b6 + 2b5 + 2b4) * q^43 + (-b9 + b8 - b7 + 3b5 - b4 + 2b3 + 2b1) q^45 + (-b9 - b8 + 2b5 + b1) q^47 - q^49 + (b9 - b8 + 2b6 - 2b4 - b2 - 3) * q^51 + (-b9 - b8 - b6 + 2b5 - b4 - 2b1) * q^53 + (-b9 + b8 - 2b7 + 2b6 + 2b5 + b1 - 2) q^55 + (b9 + b8 - 2b7 - b6 - b4 + 4b1) * q^57 + (2b9 - 2b8 + b6 - b4 - 2b3 - 2b2 + 4) * q^59 + (-b6 + b4 + 2b2 - 2) q^61 + (b6 - b5 + b4 - b1) * q^63 + (3b9 - b7 - b6 + 2b5 - 2b3 + b2 + 3b1) * q^65 + (2b6 + 2b4 - 2b1) q^67 + (2b6 - 2b4 + 2b3 - 2b2 - 2) * q^69 + (-b9 + b8 - b6 + b4 - 2b3 + 2b2) * q^71 + (-b9 - b8 - b6 - 2b5 - b4 - 2b1) * q^73 + (3b9 - b8 + b6 - 2b5 - b4 - 2b3 + b1 - 6) q^75 + (-b7 + b6 - b5 + b4) * q^77 + (b6 - b4 - 2b3 - b2 + 7) q^79 + (-b9 + b8 + b6 - b4 + 6b3 + 2b2 - 3) * q^81 + (-b9 - b8 + 4b7 - 2b6 + 6b5 - 2b4 - 2b1) q^83 + (b9 - b8 + 2b6 - 4b5 + b3 + 2b1 - 2) q^85 + (-3b9 - 3b8 - 2b5 + 3b1) * q^87 + (-2b9 + 2b8 + 2b6 - 2b4 - 2b3 - 2b2) * q^89 + (2b3 + b2 - 1) q^91 + (4b9 + 4b8 - 2b7 - 2b5 - 2b1) q^93 + (b9 - b7 + 3b4 + b3 + b2 - b1 - 6) q^95 + (b9 + b8 + b7 - 2b6 + 3b5 - 2b4) q^97 + (4b3 + 2b2 + 6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65}+ \cdots + 80 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	1120.2.g.b

Level	$1120$
Weight	$2$
Character orbit	1120.g
Analytic conductor	$8.943$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(8.94324502638\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	10.0.65174749855744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 \) x^10 + 13x^8 + 56x^6 + 97x^4 + 61x^2 + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{9} + 12\nu^{7} + 44\nu^{5} + 55\nu^{3} + 20\nu ) / 2 \) (v^9 + 12v^7 + 44v^5 + 55v^3 + 20v) / 2
\(\beta_{2}\)	\(=\)	\( ( -3\nu^{8} - 34\nu^{6} - 112\nu^{4} - 107\nu^{2} - 2 ) / 2 \) (-3v^8 - 34v^6 - 112v^4 - 107v^2 - 2) / 2
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{8} - 34\nu^{6} - 112\nu^{4} - 111\nu^{2} - 12 ) / 2 \) (-3v^8 - 34v^6 - 112v^4 - 111v^2 - 12) / 2
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{9} - 3\nu^{8} - 34\nu^{7} - 34\nu^{6} - 110\nu^{5} - 110\nu^{4} - 93\nu^{3} - 97\nu^{2} + 16\nu ) / 4 \) (-3v^9 - 3v^8 - 34v^7 - 34v^6 - 110v^5 - 110v^4 - 93v^3 - 97v^2 + 16*v) / 4
\(\beta_{5}\)	\(=\)	\( ( -2\nu^{9} - 23\nu^{7} - 78\nu^{5} - 82\nu^{3} - 13\nu ) / 2 \) (-2v^9 - 23v^7 - 78v^5 - 82v^3 - 13*v) / 2
\(\beta_{6}\)	\(=\)	\( ( -3\nu^{9} + 3\nu^{8} - 34\nu^{7} + 34\nu^{6} - 110\nu^{5} + 110\nu^{4} - 93\nu^{3} + 97\nu^{2} + 16\nu ) / 4 \) (-3v^9 + 3v^8 - 34v^7 + 34v^6 - 110v^5 + 110v^4 - 93v^3 + 97v^2 + 16*v) / 4
\(\beta_{7}\)	\(=\)	\( ( 3\nu^{9} + 35\nu^{7} + 122\nu^{5} + 137\nu^{3} + 29\nu ) / 2 \) (3v^9 + 35v^7 + 122v^5 + 137v^3 + 29*v) / 2
\(\beta_{8}\)	\(=\)	\( ( 5\nu^{9} + 5\nu^{8} + 56\nu^{7} + 58\nu^{6} + 178\nu^{5} + 198\nu^{4} + 151\nu^{3} + 203\nu^{2} - 14\nu + 20 ) / 4 \) (5v^9 + 5v^8 + 56v^7 + 58v^6 + 178v^5 + 198v^4 + 151v^3 + 203v^2 - 14*v + 20) / 4
\(\beta_{9}\)	\(=\)	\( ( 5\nu^{9} - 5\nu^{8} + 56\nu^{7} - 58\nu^{6} + 178\nu^{5} - 198\nu^{4} + 151\nu^{3} - 203\nu^{2} - 14\nu - 20 ) / 4 \) (5v^9 - 5v^8 + 56v^7 - 58v^6 + 178v^5 - 198v^4 + 151v^3 - 203v^2 - 14*v - 20) / 4

\(\nu\)	\(=\)	\( ( -\beta_{7} - \beta_{5} + \beta_1 ) / 2 \) (-b7 - b5 + b1) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{3} + \beta_{2} - 5 ) / 2 \) (-b3 + b2 - 5) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{9} + \beta_{8} + 4\beta_{7} + \beta_{6} + 6\beta_{5} + \beta_{4} - 2\beta_1 ) / 2 \) (b9 + b8 + 4b7 + b6 + 6b5 + b4 - 2*b1) / 2
\(\nu^{4}\)	\(=\)	\( ( -2\beta_{6} + 2\beta_{4} + 5\beta_{3} - 7\beta_{2} + 23 ) / 2 \) (-2b6 + 2b4 + 5b3 - 7b2 + 23) / 2
\(\nu^{5}\)	\(=\)	\( ( -8\beta_{9} - 8\beta_{8} - 21\beta_{7} - 6\beta_{6} - 41\beta_{5} - 6\beta_{4} + 3\beta_1 ) / 2 \) (-8b9 - 8b8 - 21b7 - 6b6 - 41b5 - 6b4 + 3*b1) / 2
\(\nu^{6}\)	\(=\)	\( ( -3\beta_{9} + 3\beta_{8} + 17\beta_{6} - 17\beta_{4} - 24\beta_{3} + 46\beta_{2} - 128 ) / 2 \) (-3b9 + 3b8 + 17b6 - 17b4 - 24b3 + 46b2 - 128) / 2
\(\nu^{7}\)	\(=\)	\( ( 52\beta_{9} + 52\beta_{8} + 125\beta_{7} + 32\beta_{6} + 273\beta_{5} + 32\beta_{4} + 7\beta_1 ) / 2 \) (52b9 + 52b8 + 125b7 + 32b6 + 273b5 + 32b4 + 7*b1) / 2
\(\nu^{8}\)	\(=\)	\( ( 34\beta_{9} - 34\beta_{8} - 118\beta_{6} + 118\beta_{4} + 121\beta_{3} - 297\beta_{2} + 769 ) / 2 \) (34b9 - 34b8 - 118b6 + 118b4 + 121b3 - 297b2 + 769) / 2
\(\nu^{9}\)	\(=\)	\( ( -327\beta_{9} - 327\beta_{8} - 776\beta_{7} - 175\beta_{6} - 1782\beta_{5} - 175\beta_{4} - 122\beta_1 ) / 2 \) (-327b9 - 327b8 - 776b7 - 175b6 - 1782b5 - 175b4 - 122*b1) / 2

\(n\)	\(351\)	\(421\)	\(801\)	\(897\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 449.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} \) T^10
$3$	\( T^{10} + 22 T^{8} + \cdots + 256 \) T^10 + 22T^8 + 165T^6 + 532T^4 + 708T^2 + 256
$5$	\( T^{10} + 2 T^{9} + \cdots + 3125 \) T^10 + 2T^9 - T^8 - 4T^7 + 20T^6 + 116T^5 + 100T^4 - 100T^3 - 125T^2 + 1250T + 3125
$7$	\( (T^{2} + 1)^{5} \) (T^2 + 1)^5
$11$	\( (T^{5} + 4 T^{4} + \cdots - 128)^{2} \) (T^5 + 4T^4 - 25T^3 - 152T^2 - 248T - 128)^2
$13$	\( T^{10} + 94 T^{8} + \cdots + 678976 \) T^10 + 94T^8 + 2965T^6 + 41932T^4 + 274724T^2 + 678976
$17$	\( T^{10} + 66 T^{8} + \cdots + 1024 \) T^10 + 66T^8 + 937T^6 + 3944T^4 + 4496T^2 + 1024
$19$	\( (T^{5} - 12 T^{4} + \cdots - 256)^{2} \) (T^5 - 12T^4 + 30T^3 + 72T^2 - 184T - 256)^2
$23$	\( T^{10} + 104 T^{8} + \cdots + 262144 \) T^10 + 104T^8 + 2832T^6 + 31488T^4 + 151552T^2 + 262144
$29$	\( (T^{5} - 12 T^{4} + \cdots - 4744)^{2} \) (T^5 - 12T^4 - 15T^3 + 462T^2 - 124T - 4744)^2
$31$	\( (T^{5} + 12 T^{4} + \cdots - 4096)^{2} \) (T^5 + 12T^4 - 56T^3 - 1184T^2 - 4480T - 4096)^2
$37$	\( T^{10} + 136 T^{8} + \cdots + 65536 \) T^10 + 136T^8 + 5648T^6 + 73728T^4 + 196608T^2 + 65536
$41$	\( (T^{5} + 2 T^{4} + \cdots - 128)^{2} \) (T^5 + 2T^4 - 100T^3 - 632T^2 - 1056T - 128)^2
$43$	\( T^{10} + \cdots + 228130816 \) T^10 + 344T^8 + 41040T^6 + 2136320T^4 + 45581312T^2 + 228130816
$47$	\( T^{10} + 122 T^{8} + \cdots + 4096 \) T^10 + 122T^8 + 4617T^6 + 69392T^4 + 358976T^2 + 4096
$53$	\( T^{10} + 208 T^{8} + \cdots + 3444736 \) T^10 + 208T^8 + 13920T^6 + 333312T^4 + 2001152T^2 + 3444736
$59$	\( (T^{5} - 16 T^{4} + \cdots - 58112)^{2} \) (T^5 - 16T^4 - 106T^3 + 2048T^2 + 2232T - 58112)^2
$61$	\( (T^{5} + 10 T^{4} + \cdots + 1648)^{2} \) (T^5 + 10T^4 - 46T^3 - 468T^2 + 72T + 1648)^2
$67$	\( T^{10} + 232 T^{8} + \cdots + 262144 \) T^10 + 232T^8 + 10768T^6 + 101888T^4 + 299008T^2 + 262144
$71$	\( (T^{5} + 4 T^{4} + \cdots - 1024)^{2} \) (T^5 + 4T^4 - 228T^3 - 1344T^2 - 2240T - 1024)^2
$73$	\( T^{10} + 224 T^{8} + \cdots + 4194304 \) T^10 + 224T^8 + 13792T^6 + 306688T^4 + 2052352T^2 + 4194304
$79$	\( (T^{5} - 32 T^{4} + \cdots - 3904)^{2} \) (T^5 - 32T^4 + 363T^3 - 1884T^2 + 4500T - 3904)^2
$83$	\( T^{10} + \cdots + 750321664 \) T^10 + 684T^8 + 155812T^6 + 12747648T^4 + 192013312T^2 + 750321664
$89$	\( (T^{5} + 2 T^{4} + \cdots + 3104)^{2} \) (T^5 + 2T^4 - 168T^3 - 400T^2 + 1936T + 3104)^2
$97$	\( T^{10} + 274 T^{8} + \cdots + 11723776 \) T^10 + 274T^8 + 26681T^6 + 1074792T^4 + 14907024T^2 + 11723776
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