LMFDB - Newform orbit 435.2.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,2,Mod(376,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("435.376");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 435 = 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	435.d (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:	$3.47349248793$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.1752763295712256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 17x^{8} + 96x^{6} + 201x^{4} + 121x^{2} + 16 $ x^10 + 17x^8 + 96x^6 + 201x^4 + 121x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
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^{2} + \beta_{4} q^{3} + (\beta_{2} - 1) q^{4} - q^{5} - \beta_{3} q^{6} - \beta_{5} q^{7} + (\beta_{9} - \beta_{8} + \beta_{4} - \beta_1) q^{8} - q^{9} - \beta_1 q^{10} + ( - \beta_{8} + \beta_{4}) q^{11}+ \cdots + (\beta_{8} - \beta_{4}) q^{99}+O(q^{100}) $ q + b1 * q^2 + b4 * q^3 + (b2 - 1) * q^4 - q^5 - b3 * q^6 - b5 * q^7 + (b9 - b8 + b4 - b1) * q^8 - q^9 - b1 * q^10 + (-b8 + b4) * q^11 + (-b9 - b4) * q^12 + (-b5 - 2b3) q^13 + (b9 - b8 + b6 + b4) * q^14 - b4 * q^15 + (b5 - 2b3 - b2 + 3) q^16 + (-b6 + 2b4) q^17 - b1 * q^18 + 2b4 q^19 + (-b2 + 1) * q^20 + b6 * q^21 + (b7 + b5 - b2 + 1) * q^22 + (b7 + b5 + 2b2 + 1) q^23 + (b7 + b3 + b2 - 1) * q^24 + q^25 + (-b9 - b8 + b6 - 5b4) q^26 - b4 * q^27 + (b7 - 2b3 - b2 + 1) q^28 + (b8 - b7 + b4 - 2b1) q^29 + b3 * q^30 + 2b4 q^31 + (-2b9 - b6 - 6b4 + 3b1) q^32 + (b7 - 1) * q^33 + (-b7 - b5 - 2b3 - b2 + 1) q^34 + b5 * q^35 + (-b2 + 1) * q^36 + (2b9 + b8 - b6 - b4) q^37 - 2b3 q^38 + (b6 - 2b1) q^39 + (-b9 + b8 - b4 + b1) * q^40 + (-b8 + b6 + b4 - 2b1) q^41 + (b7 + b5 + b2 - 1) * q^42 + (b8 + b6 - b4 + 2b1) q^43 + (-b9 + b8 - 2b6 + b4 + 4b1) * q^44 + q^45 + (2b9 - 2b6 + 2b4 - 2b1) * q^46 + (-b6 - 2b4 + 2b1) * q^47 + (b9 - b6 + 3b4 - 2b1) * q^48 + (-2b7 - b5 + 2b3 + 1) * q^49 + b1 * q^50 + (-b5 - 2) * q^51 + (3b7 + 4b3 + b2 - 1) * q^52 + (b7 - b5 + 2b3 - 3) q^53 + b3 * q^54 + (b8 - b4) * q^55 + (b6 - 4b4 + 4b1) * q^56 - 2 * q^57 + (-b9 - b8 - b7 + b6 - b5 - b4 - 2b3 - b2 - b1 + 5) q^58 + (-2b7 + 4b3 - 2) * q^59 + (b9 + b4) * q^60 - 2b6 q^61 - 2b3 q^62 + b5 * q^63 + (b7 + b5 + 6b3 + 2b2 - 4) * q^64 + (b5 + 2b3) q^65 + (b9 + b8 - b6 + b4) * q^66 + (-2b7 + b5 - 2) q^67 + (-3b9 - b8 - 3b4 + 2b1) q^68 + (-2b9 + b8 - b6 + b4) q^69 + (-b9 + b8 - b6 - b4) * q^70 - 8 * q^71 + (-b9 + b8 - b4 + b1) * q^72 + (-2b9 - b8 - b6 + b4) q^73 + (-4b7 - 2b5 - 4b3 - 2b2 + 2) * q^74 + b4 * q^75 + (-2b9 - 2b4) * q^76 + (2b9 + 3b6 - 2b4 + 2b1) * q^77 + (b7 + b5 - b2 + 5) * q^78 + (4b9 + 2b6 + 2b4) q^79 + (-b5 + 2b3 + b2 - 3) q^80 + q^81 + (2b7 + 2b5 - 2b2 + 6) q^82 + (-b7 - b5 - 2b2 + 7) q^83 + (b9 + b8 + b4 - 2b1) q^84 + (b6 - 2b4) q^85 + (4b2 - 8) q^86 + (-b8 - b7 + 2b3 - 1) q^87 + (-b5 + 2b2 - 10) q^88 + (-2b9 + 2b8 + b6 + 4b4 - 2b1) * q^89 + b1 * q^90 + (b5 + 2b3 + 2b2 + 6) * q^91 + (-2b7 - 6b3 - 2b2 + 12) q^92 - 2 * q^93 + (-b7 - b5 + 2b3 + b2 - 5) q^94 - 2b4 q^95 + (-b5 - 3b3 - 2b2 + 6) * q^96 + (-2b9 + b8 + 3b6 - 5b4 - 4b1) * q^97 + (b9 - 3b8 + 3b6 + 5b4 - b1) q^98 + (b8 - b4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 14 q^{4} - 10 q^{5} - 2 q^{6} - 10 q^{9} - 4 q^{13} + 30 q^{16} + 14 q^{20} + 14 q^{22} + 2 q^{23} - 12 q^{24} + 10 q^{25} + 10 q^{28} + 2 q^{30} - 10 q^{33} + 10 q^{34} + 14 q^{36} - 4 q^{38} - 14 q^{42}+ \cdots + 62 q^{96}+O(q^{100}) $ 10 * q - 14 * q^4 - 10 * q^5 - 2 * q^6 - 10 * q^9 - 4 * q^13 + 30 * q^16 + 14 * q^20 + 14 * q^22 + 2 * q^23 - 12 * q^24 + 10 * q^25 + 10 * q^28 + 2 * q^30 - 10 * q^33 + 10 * q^34 + 14 * q^36 - 4 * q^38 - 14 * q^42 + 10 * q^45 + 14 * q^49 - 20 * q^51 - 6 * q^52 - 26 * q^53 + 2 * q^54 - 20 * q^57 + 50 * q^58 - 12 * q^59 - 4 * q^62 - 36 * q^64 + 4 * q^65 - 20 * q^67 - 80 * q^71 + 20 * q^74 + 54 * q^78 - 30 * q^80 + 10 * q^81 + 68 * q^82 + 78 * q^83 - 96 * q^86 - 6 * q^87 - 108 * q^88 + 56 * q^91 + 116 * q^92 - 20 * q^93 - 50 * q^94 + 62 * q^96

Basis of coefficient ring in terms of a root $\nu$ of $ x^{10} + 17x^{8} + 96x^{6} + 201x^{4} + 121x^{2} + 16 $ x^10 + 17*x^8 + 96*x^6 + 201*x^4 + 121*x^2 + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 3 $ v^2 + 3
$\beta_{3}$	$=$	$ ( \nu^{8} + 20\nu^{6} + 116\nu^{4} + 189\nu^{2} + 48 ) / 40 $ (v^8 + 20v^6 + 116v^4 + 189*v^2 + 48) / 40
$\beta_{4}$	$=$	$ ( 3\nu^{9} + 50\nu^{7} + 268\nu^{5} + 487\nu^{3} + 174\nu ) / 40 $ (3v^9 + 50v^7 + 268v^5 + 487v^3 + 174*v) / 40
$\beta_{5}$	$=$	$ ( \nu^{8} + 20\nu^{6} + 136\nu^{4} + 329\nu^{2} + 128 ) / 20 $ (v^8 + 20v^6 + 136v^4 + 329*v^2 + 128) / 20
$\beta_{6}$	$=$	$ ( -\nu^{9} - 20\nu^{7} - 136\nu^{5} - 349\nu^{3} - 228\nu ) / 20 $ (-v^9 - 20v^7 - 136v^5 - 349v^3 - 228v) / 20
$\beta_{7}$	$=$	$ ( -\nu^{8} - 15\nu^{6} - 71\nu^{4} - 114\nu^{2} - 38 ) / 5 $ (-v^8 - 15v^6 - 71v^4 - 114*v^2 - 38) / 5
$\beta_{8}$	$=$	$ ( -\nu^{9} - 16\nu^{7} - 84\nu^{5} - 165\nu^{3} - 100\nu ) / 8 $ (-v^9 - 16v^7 - 84v^5 - 165v^3 - 100v) / 8
$\beta_{9}$	$=$	$ ( -4\nu^{9} - 65\nu^{7} - 344\nu^{5} - 636\nu^{3} - 237\nu ) / 20 $ (-4v^9 - 65v^7 - 344v^5 - 636v^3 - 237*v) / 20

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 3 $ b2 - 3
$\nu^{3}$	$=$	$ \beta_{9} - \beta_{8} + \beta_{4} - 5\beta_1 $ b9 - b8 + b4 - 5*b1
$\nu^{4}$	$=$	$ \beta_{5} - 2\beta_{3} - 7\beta_{2} + 17 $ b5 - 2b3 - 7b2 + 17
$\nu^{5}$	$=$	$ -10\beta_{9} + 8\beta_{8} - \beta_{6} - 14\beta_{4} + 31\beta_1 $ -10b9 + 8b8 - b6 - 14b4 + 31b1
$\nu^{6}$	$=$	$ \beta_{7} - 9\beta_{5} + 26\beta_{3} + 48\beta_{2} - 110 $ b7 - 9b5 + 26b3 + 48*b2 - 110
$\nu^{7}$	$=$	$ 84\beta_{9} - 56\beta_{8} + 8\beta_{6} + 136\beta_{4} - 205\beta_1 $ 84b9 - 56b8 + 8b6 + 136b4 - 205*b1
$\nu^{8}$	$=$	$ -20\beta_{7} + 64\beta_{5} - 248\beta_{3} - 337\beta_{2} + 747 $ -20b7 + 64b5 - 248b3 - 337b2 + 747
$\nu^{9}$	$=$	$ -669\beta_{9} + 381\beta_{8} - 44\beta_{6} - 1165\beta_{4} + 1401\beta_1 $ -669b9 + 381b8 - 44b6 - 1165b4 + 1401*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/435\mathbb{Z}\right)^\times$.

$n$	$31$	$146$	$262$
$\chi(n)$	$-1$	$1$	$1$

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} $

376.1

	−	2.72674i
	−	2.37667i
	−	1.74703i
	−	0.825137i
	−	0.428173i
		0.428173i
		0.825137i
		1.74703i
		2.37667i
		2.72674i

−

2.72674i

1.00000i

−5.43512

−1.00000

2.72674

−1.78170

9.36669i

−1.00000

2.72674i

376.2

−

2.37667i

−

1.00000i

−3.64857

−1.00000

−2.37667

−1.11968

3.91810i

−1.00000

2.37667i

376.3

−

1.74703i

−

1.00000i

−1.05213

−1.00000

−1.74703

4.55534

−

1.65596i

−1.00000

1.74703i

376.4

−

0.825137i

1.00000i

1.31915

−1.00000

0.825137

1.95267

−

2.73875i

−1.00000

0.825137i

376.5

−

0.428173i

−

1.00000i

1.81667

−1.00000

−0.428173

−3.60663

−

1.63419i

−1.00000

0.428173i

376.6

0.428173i

1.00000i

1.81667

−1.00000

−0.428173

−3.60663

1.63419i

−1.00000

−

0.428173i

376.7

0.825137i

−

1.00000i

1.31915

−1.00000

0.825137

1.95267

2.73875i

−1.00000

−

0.825137i

376.8

1.74703i

1.00000i

−1.05213

−1.00000

−1.74703

4.55534

1.65596i

−1.00000

−

1.74703i

376.9

2.37667i

1.00000i

−3.64857

−1.00000

−2.37667

−1.11968

−

3.91810i

−1.00000

−

2.37667i

376.10

2.72674i

−

1.00000i

−5.43512

−1.00000

2.72674

−1.78170

−

9.36669i

−1.00000

−

2.72674i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	435.2.d.a	✓	10
3.b	odd	2	1		1305.2.d.d		10
5.b	even	2	1		2175.2.d.f		10
5.c	odd	4	1		2175.2.f.d		10
5.c	odd	4	1		2175.2.f.e		10
29.b	even	2	1	inner	435.2.d.a	✓	10
87.d	odd	2	1		1305.2.d.d		10
145.d	even	2	1		2175.2.d.f		10
145.h	odd	4	1		2175.2.f.d		10
145.h	odd	4	1		2175.2.f.e		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
435.2.d.a	✓	10	1.a	even	1	1	trivial
435.2.d.a	✓	10	29.b	even	2	1	inner
1305.2.d.d		10	3.b	odd	2	1
1305.2.d.d		10	87.d	odd	2	1
2175.2.d.f		10	5.b	even	2	1
2175.2.d.f		10	145.d	even	2	1
2175.2.f.d		10	5.c	odd	4	1
2175.2.f.d		10	145.h	odd	4	1
2175.2.f.e		10	5.c	odd	4	1
2175.2.f.e		10	145.h	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} + 17T_{2}^{8} + 96T_{2}^{6} + 201T_{2}^{4} + 121T_{2}^{2} + 16 $ T2^10 + 17*T2^8 + 96*T2^6 + 201*T2^4 + 121*T2^2 + 16 acting on $S_{2}^{\mathrm{new}}(435, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} + 17 T^{8} + \cdots + 16 $ T^10 + 17T^8 + 96T^6 + 201T^4 + 121T^2 + 16
$3$	$ (T^{2} + 1)^{5} $ (T^2 + 1)^5
$5$	$ (T + 1)^{10} $ (T + 1)^10
$7$	$ (T^{5} - 21 T^{3} + \cdots + 64)^{2} $ (T^5 - 21T^3 - 16T^2 + 64*T + 64)^2
$11$	$ T^{10} + 59 T^{8} + \cdots + 35344 $ T^10 + 59T^8 + 1227T^6 + 10745T^4 + 36728T^2 + 35344
$13$	$ (T^{5} + 2 T^{4} + \cdots - 368)^{2} $ (T^5 + 2T^4 - 39T^3 - 16T^2 + 404T - 368)^2
$17$	$ T^{10} + 62 T^{8} + \cdots + 64 $ T^10 + 62T^8 + 1257T^6 + 10220T^4 + 28592T^2 + 64
$19$	$ (T^{2} + 4)^{5} $ (T^2 + 4)^5
$23$	$ (T^{5} - T^{4} - 84 T^{3} + \cdots - 3328)^{2} $ (T^5 - T^4 - 84T^3 + 128T^2 + 1728*T - 3328)^2
$29$	$ T^{10} + 37 T^{8} + \cdots + 20511149 $ T^10 + 37T^8 + 208T^7 + 806T^6 + 7968T^5 + 23374T^4 + 174928T^3 + 902393*T^2 + 20511149
$31$	$ (T^{2} + 4)^{5} $ (T^2 + 4)^5
$37$	$ T^{10} + \cdots + 666052864 $ T^10 + 321T^8 + 39248T^6 + 2288096T^4 + 63466496T^2 + 666052864
$41$	$ T^{10} + 161 T^{8} + \cdots + 65536 $ T^10 + 161T^8 + 7032T^6 + 103440T^4 + 396288T^2 + 65536
$43$	$ T^{10} + 233 T^{8} + \cdots + 16777216 $ T^10 + 233T^8 + 19360T^6 + 676096T^4 + 8519680T^2 + 16777216
$47$	$ T^{10} + 94 T^{8} + \cdots + 861184 $ T^10 + 94T^8 + 3233T^6 + 49924T^4 + 345472T^2 + 861184
$53$	$ (T^{5} + 13 T^{4} + \cdots + 784)^{2} $ (T^5 + 13T^4 - 48T^3 - 632T^2 + 1456T + 784)^2
$59$	$ (T^{5} + 6 T^{4} + \cdots + 21376)^{2} $ (T^5 + 6T^4 - 164T^3 - 792T^2 + 6592T + 21376)^2
$61$	$ T^{10} + 168 T^{8} + \cdots + 4194304 $ T^10 + 168T^8 + 9104T^6 + 188416T^4 + 1572864T^2 + 4194304
$67$	$ (T^{5} + 10 T^{4} + \cdots + 3616)^{2} $ (T^5 + 10T^4 - 97T^3 - 902T^2 + 664T + 3616)^2
$71$	$ (T + 8)^{10} $ (T + 8)^10
$73$	$ T^{10} + 297 T^{8} + \cdots + 1183744 $ T^10 + 297T^8 + 21496T^6 + 411152T^4 + 1333760T^2 + 1183744
$79$	$ T^{10} + \cdots + 11097358336 $ T^10 + 780T^8 + 219056T^6 + 26443840T^4 + 1249794048T^2 + 11097358336
$83$	$ (T^{5} - 39 T^{4} + \cdots - 4352)^{2} $ (T^5 - 39T^4 + 524T^3 - 2848T^2 + 6080T - 4352)^2
$89$	$ T^{10} + 310 T^{8} + \cdots + 16384 $ T^10 + 310T^8 + 8713T^6 + 82724T^4 + 256768T^2 + 16384
$97$	$ T^{10} + \cdots + 536015104 $ T^10 + 785T^8 + 213424T^6 + 23521888T^4 + 901434880T^2 + 536015104
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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:	\( N \)	\(=\)	\( 435 = 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	435.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 1) q^{4} - q^{5} - \beta_{3} q^{6} - \beta_{5} q^{7} + (\beta_{9} - \beta_{8} + \beta_{4} - \beta_1) q^{8} - q^{9} - \beta_1 q^{10} + ( - \beta_{8} + \beta_{4}) q^{11}+ \cdots + (\beta_{8} - \beta_{4}) q^{99}+O(q^{100}) \) q + b1 * q^2 + b4 * q^3 + (b2 - 1) * q^4 - q^5 - b3 * q^6 - b5 * q^7 + (b9 - b8 + b4 - b1) * q^8 - q^9 - b1 * q^10 + (-b8 + b4) * q^11 + (-b9 - b4) * q^12 + (-b5 - 2b3) q^13 + (b9 - b8 + b6 + b4) * q^14 - b4 * q^15 + (b5 - 2b3 - b2 + 3) q^16 + (-b6 + 2b4) q^17 - b1 * q^18 + 2b4 q^19 + (-b2 + 1) * q^20 + b6 * q^21 + (b7 + b5 - b2 + 1) * q^22 + (b7 + b5 + 2b2 + 1) q^23 + (b7 + b3 + b2 - 1) * q^24 + q^25 + (-b9 - b8 + b6 - 5b4) q^26 - b4 * q^27 + (b7 - 2b3 - b2 + 1) q^28 + (b8 - b7 + b4 - 2b1) q^29 + b3 * q^30 + 2b4 q^31 + (-2b9 - b6 - 6b4 + 3b1) q^32 + (b7 - 1) * q^33 + (-b7 - b5 - 2b3 - b2 + 1) q^34 + b5 * q^35 + (-b2 + 1) * q^36 + (2b9 + b8 - b6 - b4) q^37 - 2b3 q^38 + (b6 - 2b1) q^39 + (-b9 + b8 - b4 + b1) * q^40 + (-b8 + b6 + b4 - 2b1) q^41 + (b7 + b5 + b2 - 1) * q^42 + (b8 + b6 - b4 + 2b1) q^43 + (-b9 + b8 - 2b6 + b4 + 4b1) * q^44 + q^45 + (2b9 - 2b6 + 2b4 - 2b1) * q^46 + (-b6 - 2b4 + 2b1) * q^47 + (b9 - b6 + 3b4 - 2b1) * q^48 + (-2b7 - b5 + 2b3 + 1) * q^49 + b1 * q^50 + (-b5 - 2) * q^51 + (3b7 + 4b3 + b2 - 1) * q^52 + (b7 - b5 + 2b3 - 3) q^53 + b3 * q^54 + (b8 - b4) * q^55 + (b6 - 4b4 + 4b1) * q^56 - 2 * q^57 + (-b9 - b8 - b7 + b6 - b5 - b4 - 2b3 - b2 - b1 + 5) q^58 + (-2b7 + 4b3 - 2) * q^59 + (b9 + b4) * q^60 - 2b6 q^61 - 2b3 q^62 + b5 * q^63 + (b7 + b5 + 6b3 + 2b2 - 4) * q^64 + (b5 + 2b3) q^65 + (b9 + b8 - b6 + b4) * q^66 + (-2b7 + b5 - 2) q^67 + (-3b9 - b8 - 3b4 + 2b1) q^68 + (-2b9 + b8 - b6 + b4) q^69 + (-b9 + b8 - b6 - b4) * q^70 - 8 * q^71 + (-b9 + b8 - b4 + b1) * q^72 + (-2b9 - b8 - b6 + b4) q^73 + (-4b7 - 2b5 - 4b3 - 2b2 + 2) * q^74 + b4 * q^75 + (-2b9 - 2b4) * q^76 + (2b9 + 3b6 - 2b4 + 2b1) * q^77 + (b7 + b5 - b2 + 5) * q^78 + (4b9 + 2b6 + 2b4) q^79 + (-b5 + 2b3 + b2 - 3) q^80 + q^81 + (2b7 + 2b5 - 2b2 + 6) q^82 + (-b7 - b5 - 2b2 + 7) q^83 + (b9 + b8 + b4 - 2b1) q^84 + (b6 - 2b4) q^85 + (4b2 - 8) q^86 + (-b8 - b7 + 2b3 - 1) q^87 + (-b5 + 2b2 - 10) q^88 + (-2b9 + 2b8 + b6 + 4b4 - 2b1) * q^89 + b1 * q^90 + (b5 + 2b3 + 2b2 + 6) * q^91 + (-2b7 - 6b3 - 2b2 + 12) q^92 - 2 * q^93 + (-b7 - b5 + 2b3 + b2 - 5) q^94 - 2b4 q^95 + (-b5 - 3b3 - 2b2 + 6) * q^96 + (-2b9 + b8 + 3b6 - 5b4 - 4b1) * q^97 + (b9 - 3b8 + 3b6 + 5b4 - b1) q^98 + (b8 - b4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 14 q^{4} - 10 q^{5} - 2 q^{6} - 10 q^{9} - 4 q^{13} + 30 q^{16} + 14 q^{20} + 14 q^{22} + 2 q^{23} - 12 q^{24} + 10 q^{25} + 10 q^{28} + 2 q^{30} - 10 q^{33} + 10 q^{34} + 14 q^{36} - 4 q^{38} - 14 q^{42}+ \cdots + 62 q^{96}+O(q^{100}) \) 10 * q - 14 * q^4 - 10 * q^5 - 2 * q^6 - 10 * q^9 - 4 * q^13 + 30 * q^16 + 14 * q^20 + 14 * q^22 + 2 * q^23 - 12 * q^24 + 10 * q^25 + 10 * q^28 + 2 * q^30 - 10 * q^33 + 10 * q^34 + 14 * q^36 - 4 * q^38 - 14 * q^42 + 10 * q^45 + 14 * q^49 - 20 * q^51 - 6 * q^52 - 26 * q^53 + 2 * q^54 - 20 * q^57 + 50 * q^58 - 12 * q^59 - 4 * q^62 - 36 * q^64 + 4 * q^65 - 20 * q^67 - 80 * q^71 + 20 * q^74 + 54 * q^78 - 30 * q^80 + 10 * q^81 + 68 * q^82 + 78 * q^83 - 96 * q^86 - 6 * q^87 - 108 * q^88 + 56 * q^91 + 116 * q^92 - 20 * q^93 - 50 * q^94 + 62 * q^96

Introduction

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Newspace parameters

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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	435.2.d.a

Level	$435$
Weight	$2$
Character orbit	435.d
Analytic conductor	$3.473$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.47349248793\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	10.0.1752763295712256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} + 17x^{8} + 96x^{6} + 201x^{4} + 121x^{2} + 16 \) x^10 + 17x^8 + 96x^6 + 201x^4 + 121x^2 + 16
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 3 \) v^2 + 3
\(\beta_{3}\)	\(=\)	\( ( \nu^{8} + 20\nu^{6} + 116\nu^{4} + 189\nu^{2} + 48 ) / 40 \) (v^8 + 20v^6 + 116v^4 + 189*v^2 + 48) / 40
\(\beta_{4}\)	\(=\)	\( ( 3\nu^{9} + 50\nu^{7} + 268\nu^{5} + 487\nu^{3} + 174\nu ) / 40 \) (3v^9 + 50v^7 + 268v^5 + 487v^3 + 174*v) / 40
\(\beta_{5}\)	\(=\)	\( ( \nu^{8} + 20\nu^{6} + 136\nu^{4} + 329\nu^{2} + 128 ) / 20 \) (v^8 + 20v^6 + 136v^4 + 329*v^2 + 128) / 20
\(\beta_{6}\)	\(=\)	\( ( -\nu^{9} - 20\nu^{7} - 136\nu^{5} - 349\nu^{3} - 228\nu ) / 20 \) (-v^9 - 20v^7 - 136v^5 - 349v^3 - 228v) / 20
\(\beta_{7}\)	\(=\)	\( ( -\nu^{8} - 15\nu^{6} - 71\nu^{4} - 114\nu^{2} - 38 ) / 5 \) (-v^8 - 15v^6 - 71v^4 - 114*v^2 - 38) / 5
\(\beta_{8}\)	\(=\)	\( ( -\nu^{9} - 16\nu^{7} - 84\nu^{5} - 165\nu^{3} - 100\nu ) / 8 \) (-v^9 - 16v^7 - 84v^5 - 165v^3 - 100v) / 8
\(\beta_{9}\)	\(=\)	\( ( -4\nu^{9} - 65\nu^{7} - 344\nu^{5} - 636\nu^{3} - 237\nu ) / 20 \) (-4v^9 - 65v^7 - 344v^5 - 636v^3 - 237*v) / 20

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} - 3 \) b2 - 3
\(\nu^{3}\)	\(=\)	\( \beta_{9} - \beta_{8} + \beta_{4} - 5\beta_1 \) b9 - b8 + b4 - 5*b1
\(\nu^{4}\)	\(=\)	\( \beta_{5} - 2\beta_{3} - 7\beta_{2} + 17 \) b5 - 2b3 - 7b2 + 17
\(\nu^{5}\)	\(=\)	\( -10\beta_{9} + 8\beta_{8} - \beta_{6} - 14\beta_{4} + 31\beta_1 \) -10b9 + 8b8 - b6 - 14b4 + 31b1
\(\nu^{6}\)	\(=\)	\( \beta_{7} - 9\beta_{5} + 26\beta_{3} + 48\beta_{2} - 110 \) b7 - 9b5 + 26b3 + 48*b2 - 110
\(\nu^{7}\)	\(=\)	\( 84\beta_{9} - 56\beta_{8} + 8\beta_{6} + 136\beta_{4} - 205\beta_1 \) 84b9 - 56b8 + 8b6 + 136b4 - 205*b1
\(\nu^{8}\)	\(=\)	\( -20\beta_{7} + 64\beta_{5} - 248\beta_{3} - 337\beta_{2} + 747 \) -20b7 + 64b5 - 248b3 - 337b2 + 747
\(\nu^{9}\)	\(=\)	\( -669\beta_{9} + 381\beta_{8} - 44\beta_{6} - 1165\beta_{4} + 1401\beta_1 \) -669b9 + 381b8 - 44b6 - 1165b4 + 1401*b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 376.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} + 17 T^{8} + \cdots + 16 \) T^10 + 17T^8 + 96T^6 + 201T^4 + 121T^2 + 16
$3$	\( (T^{2} + 1)^{5} \) (T^2 + 1)^5
$5$	\( (T + 1)^{10} \) (T + 1)^10
$7$	\( (T^{5} - 21 T^{3} + \cdots + 64)^{2} \) (T^5 - 21T^3 - 16T^2 + 64*T + 64)^2
$11$	\( T^{10} + 59 T^{8} + \cdots + 35344 \) T^10 + 59T^8 + 1227T^6 + 10745T^4 + 36728T^2 + 35344
$13$	\( (T^{5} + 2 T^{4} + \cdots - 368)^{2} \) (T^5 + 2T^4 - 39T^3 - 16T^2 + 404T - 368)^2
$17$	\( T^{10} + 62 T^{8} + \cdots + 64 \) T^10 + 62T^8 + 1257T^6 + 10220T^4 + 28592T^2 + 64
$19$	\( (T^{2} + 4)^{5} \) (T^2 + 4)^5
$23$	\( (T^{5} - T^{4} - 84 T^{3} + \cdots - 3328)^{2} \) (T^5 - T^4 - 84T^3 + 128T^2 + 1728*T - 3328)^2
$29$	\( T^{10} + 37 T^{8} + \cdots + 20511149 \) T^10 + 37T^8 + 208T^7 + 806T^6 + 7968T^5 + 23374T^4 + 174928T^3 + 902393*T^2 + 20511149
$31$	\( (T^{2} + 4)^{5} \) (T^2 + 4)^5
$37$	\( T^{10} + \cdots + 666052864 \) T^10 + 321T^8 + 39248T^6 + 2288096T^4 + 63466496T^2 + 666052864
$41$	\( T^{10} + 161 T^{8} + \cdots + 65536 \) T^10 + 161T^8 + 7032T^6 + 103440T^4 + 396288T^2 + 65536
$43$	\( T^{10} + 233 T^{8} + \cdots + 16777216 \) T^10 + 233T^8 + 19360T^6 + 676096T^4 + 8519680T^2 + 16777216
$47$	\( T^{10} + 94 T^{8} + \cdots + 861184 \) T^10 + 94T^8 + 3233T^6 + 49924T^4 + 345472T^2 + 861184
$53$	\( (T^{5} + 13 T^{4} + \cdots + 784)^{2} \) (T^5 + 13T^4 - 48T^3 - 632T^2 + 1456T + 784)^2
$59$	\( (T^{5} + 6 T^{4} + \cdots + 21376)^{2} \) (T^5 + 6T^4 - 164T^3 - 792T^2 + 6592T + 21376)^2
$61$	\( T^{10} + 168 T^{8} + \cdots + 4194304 \) T^10 + 168T^8 + 9104T^6 + 188416T^4 + 1572864T^2 + 4194304
$67$	\( (T^{5} + 10 T^{4} + \cdots + 3616)^{2} \) (T^5 + 10T^4 - 97T^3 - 902T^2 + 664T + 3616)^2
$71$	\( (T + 8)^{10} \) (T + 8)^10
$73$	\( T^{10} + 297 T^{8} + \cdots + 1183744 \) T^10 + 297T^8 + 21496T^6 + 411152T^4 + 1333760T^2 + 1183744
$79$	\( T^{10} + \cdots + 11097358336 \) T^10 + 780T^8 + 219056T^6 + 26443840T^4 + 1249794048T^2 + 11097358336
$83$	\( (T^{5} - 39 T^{4} + \cdots - 4352)^{2} \) (T^5 - 39T^4 + 524T^3 - 2848T^2 + 6080T - 4352)^2
$89$	\( T^{10} + 310 T^{8} + \cdots + 16384 \) T^10 + 310T^8 + 8713T^6 + 82724T^4 + 256768T^2 + 16384
$97$	\( T^{10} + \cdots + 536015104 \) T^10 + 785T^8 + 213424T^6 + 23521888T^4 + 901434880T^2 + 536015104
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\(n\)	\(31\)	\(146\)	\(262\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)