LMFDB - Newform orbit 603.2.u.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [603,2,Mod(64,603)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(603, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("603.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 603 = 3^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	603.u (of order $11$, degree $10$, 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:	$4.81497924188$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 67)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/603\mathbb{Z}\right)^\times$.

$n$	$136$	$470$
$\chi(n)$	$\zeta_{22}^{4}$	$1$

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} $

64.1

0.142315	−	0.989821i
0.959493	−	0.281733i
−0.841254	+	0.540641i
0.654861	−	0.755750i
−0.415415	−	0.909632i
0.142315	+	0.989821i
−0.415415	+	0.909632i
0.959493	+	0.281733i
−0.841254	−	0.540641i
0.654861	+	0.755750i

2.13843

0.627899i

2.49611

1.60416i

−0.381761

−

0.835939i

2.87102

0.843008i

1.41153

1.62899i

−0.291482

−

2.02730i

82.1

2.01722

−

1.29639i

1.55773

−

3.41095i

−1.75667

−

2.02730i

−0.938384

0.603063i

−0.597131

−

4.15314i

−6.17177

−

1.81219i

91.1

−0.0366213

0.0801894i

1.30463

1.50563i

−0.260554

1.81219i

−0.253098

0.554206i

−0.337683

0.0991526i

−0.135777

−

0.0872586i

226.1

−0.220047

−

1.53046i

−0.374908

0.110083i

0.601808

−

0.386758i

−0.391340

−

2.72183i

−1.03365

−

2.26339i

−0.724345

−

0.835939i

397.1

−0.898983

1.03748i

0.0164316

0.114284i

0.297176

−

0.0872586i

−1.28820

1.48666i

−2.44306

−

1.57006i

−0.176627

0.386758i

424.1

2.13843

−

0.627899i

2.49611

−

1.60416i

−0.381761

0.835939i

2.87102

−

0.843008i

1.41153

−

1.62899i

−0.291482

2.02730i

442.1

−0.898983

−

1.03748i

0.0164316

−

0.114284i

0.297176

0.0872586i

−1.28820

−

1.48666i

−2.44306

1.57006i

−0.176627

−

0.386758i

478.1

2.01722

1.29639i

1.55773

3.41095i

−1.75667

2.02730i

−0.938384

−

0.603063i

−0.597131

4.15314i

−6.17177

1.81219i

550.1

−0.0366213

−

0.0801894i

1.30463

−

1.50563i

−0.260554

−

1.81219i

−0.253098

−

0.554206i

−0.337683

−

0.0991526i

−0.135777

0.0872586i

595.1

−0.220047

1.53046i

−0.374908

−

0.110083i

0.601808

0.386758i

−0.391340

2.72183i

−1.03365

2.26339i

−0.724345

0.835939i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
67.e	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	603.2.u.b		10
3.b	odd	2	1		67.2.e.a	✓	10
67.e	even	11	1	inner	603.2.u.b		10
201.j	even	22	1		4489.2.a.k		5
201.k	odd	22	1		67.2.e.a	✓	10
201.k	odd	22	1		4489.2.a.f		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
67.2.e.a	✓	10	3.b	odd	2	1
67.2.e.a	✓	10	201.k	odd	22	1
603.2.u.b		10	1.a	even	1	1	trivial
603.2.u.b		10	67.e	even	11	1	inner
4489.2.a.f		5	201.k	odd	22	1
4489.2.a.k		5	201.j	even	22	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} - 6T_{2}^{9} + 14T_{2}^{8} - 18T_{2}^{7} + 31T_{2}^{6} - 54T_{2}^{5} + 38T_{2}^{4} - 52T_{2}^{3} + 125T_{2}^{2} + 9T_{2} + 1 $ T2^10 - 6*T2^9 + 14*T2^8 - 18*T2^7 + 31*T2^6 - 54*T2^5 + 38*T2^4 - 52*T2^3 + 125*T2^2 + 9*T2 + 1 acting on $S_{2}^{\mathrm{new}}(603, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - 6 T^{9} + \cdots + 1 $ T^10 - 6T^9 + 14T^8 - 18T^7 + 31T^6 - 54T^5 + 38T^4 - 52T^3 + 125T^2 + 9*T + 1
$3$	$ T^{10} $ T^10
$5$	$ T^{10} + 3 T^{9} + \cdots + 1 $ T^10 + 3T^9 + 9T^8 + 5T^7 + 15T^6 - 21T^5 + 14T^4 - 13T^3 + 16T^2 - 7*T + 1
$7$	$ T^{10} - 22 T^{7} + \cdots + 121 $ T^10 - 22T^7 - 33T^6 + 22T^5 + 363T^4 + 726T^3 + 726T^2 + 363*T + 121
$11$	$ T^{10} + 15 T^{9} + \cdots + 69169 $ T^10 + 15T^9 + 137T^8 + 845T^7 + 3765T^6 + 11661T^5 + 25821T^4 + 44313T^3 + 81541T^2 + 114931*T + 69169
$13$	$ T^{10} + 18 T^{9} + \cdots + 69169 $ T^10 + 18T^9 + 159T^8 + 1014T^7 + 5184T^6 + 19546T^5 + 50065T^4 + 75367T^3 + 56307T^2 + 23933*T + 69169
$17$	$ T^{10} - 26 T^{9} + \cdots + 737881 $ T^10 - 26T^9 + 390T^8 - 3837T^7 + 27283T^6 - 142473T^5 + 557154T^4 - 1557825T^3 + 2902513T^2 - 2430970*T + 737881
$19$	$ T^{10} + T^{9} + \cdots + 1849 $ T^10 + T^9 + 45T^8 + 45T^7 + 441T^6 - 3816T^5 + 3752T^4 - 4454T^3 + 31901T^2 - 12814T + 1849
$23$	$ T^{10} - 6 T^{9} + \cdots + 1067089 $ T^10 - 6T^9 + 69T^8 - 315T^7 + 1439T^6 - 2496T^5 - 4318T^4 - 10986T^3 + 236130T^2 - 816070*T + 1067089
$29$	$ (T^{5} + 14 T^{4} + \cdots + 199)^{2} $ (T^5 + 14T^4 + 41T^3 - 125T^2 - 483T + 199)^2
$31$	$ T^{10} + 10 T^{9} + \cdots + 2374681 $ T^10 + 10T^9 + 133T^8 + 505T^7 + 1618T^6 - 18536T^5 - 46925T^4 - 149051T^3 + 1214632T^2 + 2947933*T + 2374681
$37$	$ (T^{5} + 11 T^{4} + \cdots - 979)^{2} $ (T^5 + 11T^4 - 55T^3 - 451T^2 + 1804T - 979)^2
$41$	$ T^{10} + 9 T^{9} + \cdots + 59049 $ T^10 + 9T^9 + 81T^8 + 432T^7 + 324T^6 + 243T^5 + 18225T^4 - 4374T^3 + 32805T^2 + 78732*T + 59049
$43$	$ T^{10} - 21 T^{9} + \cdots + 83375161 $ T^10 - 21T^9 + 331T^8 - 3629T^7 + 23904T^6 - 132659T^5 + 922076T^4 - 6071735T^3 + 38549358T^2 - 89082036*T + 83375161
$47$	$ T^{10} - 22 T^{9} + \cdots + 33860761 $ T^10 - 22T^9 + 297T^8 - 2937T^7 + 31482T^6 - 273317T^5 + 1422234T^4 - 4945391T^3 + 16258286T^2 - 33860761*T + 33860761
$53$	$ T^{10} + 20 T^{9} + \cdots + 21132409 $ T^10 + 20T^9 + 169T^8 + 1048T^7 + 9410T^6 + 91730T^5 + 649229T^4 + 2988781T^3 + 8505269T^2 + 13427837*T + 21132409
$59$	$ T^{10} - 17 T^{9} + \cdots + 8826841 $ T^10 - 17T^9 + 300T^8 - 1360T^7 + 15970T^6 + 48665T^5 + 767673T^4 + 6113352T^3 + 22905920T^2 + 25093066*T + 8826841
$61$	$ T^{10} - 13 T^{9} + \cdots + 5139289 $ T^10 - 13T^9 + 235T^8 - 3594T^7 + 37383T^6 - 370655T^5 + 2561975T^4 - 7747945T^3 + 13119670T^2 - 12008299*T + 5139289
$67$	$ T^{10} + \cdots + 1350125107 $ T^10 - 23T^9 + 111T^8 + 967T^7 - 6379T^6 - 13685T^5 - 427393T^4 + 4340863T^3 + 33384693T^2 - 463475783*T + 1350125107
$71$	$ T^{10} + T^{9} + \cdots + 896809 $ T^10 + T^9 - 10T^8 + 1101T^7 + 18393T^6 + 18976T^5 + 209870T^4 + 262087T^3 + 898789T^2 + 1636416T + 896809
$73$	$ T^{10} + \cdots + 1786921984 $ T^10 + 18T^9 + 280T^8 + 4424T^7 + 49360T^6 + 464672T^5 + 3407936T^4 + 8718848T^3 - 18483456T^2 - 309092864*T + 1786921984
$79$	$ T^{10} - 5 T^{9} + \cdots + 4012009 $ T^10 - 5T^9 + 3T^8 - 1676T^7 + 5366T^6 + 55175T^5 + 770775T^4 + 3219136T^3 + 12377919T^2 + 11373034*T + 4012009
$83$	$ T^{10} - 12 T^{9} + \cdots + 20529961 $ T^10 - 12T^9 - 32T^8 + 1374T^7 + 7129T^6 + 21196T^5 + 143298T^4 + 166000T^3 + 2855183T^2 + 2895309*T + 20529961
$89$	$ T^{10} + \cdots + 375623161 $ T^10 + 51T^9 + 1215T^8 + 18207T^7 + 193075T^6 + 1515777T^5 + 9035679T^4 + 40680371T^3 + 135597163T^2 + 306200419*T + 375623161
$97$	$ (T^{5} + 5 T^{4} + \cdots - 989)^{2} $ (T^5 + 5T^4 - 89T^3 - 287T^2 + 1490T - 989)^2
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Overview	Random
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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:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.u (of order \(11\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{22}^{8} - \zeta_{22}^{7} + \cdots + 1) q^{2}+ \cdots + (\zeta_{22}^{9} + \zeta_{22}^{8} + \cdots - 1) q^{8}+O(q^{10}) \) q + (z^8 - z^7 + z^4 + z^2 + 1) * q^2 + (z^8 + z^5 + z^4 + z^3 + 1) * q^4 + (z^9 + z^7 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^5 + (z^9 + z^8 - z^7 - z^6) * q^7 + (z^9 + z^8 + z^7 - z^6 + z^5 + z^4 + z^3 + z - 1) * q^8	\( q + (\zeta_{22}^{8} - \zeta_{22}^{7} + \cdots + 1) q^{2}+ \cdots + ( - 4 \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots + 2) q^{98}+O(q^{100}) \) q + (z^8 - z^7 + z^4 + z^2 + 1) * q^2 + (z^8 + z^5 + z^4 + z^3 + 1) * q^4 + (z^9 + z^7 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^5 + (z^9 + z^8 - z^7 - z^6) * q^7 + (z^9 + z^8 + z^7 - z^6 + z^5 + z^4 + z^3 + z - 1) * q^8 + (3z^9 - z^8 + 2z^7 - z^6 + 2z^5 - 2z^4 + z^3 - 2z^2 + z - 3) q^10 + (z^8 + 2z^5 + 2z^3 - 2z^2 - 2) q^11 + (-z^9 + 3z^8 - z^7 + 3z^6 - 2z^5 + 3z^4 - z^3 + 3z^2 - z) q^13 + (-z^7 - 2z^6 + 2z^4 - 2z^2 - z) q^14 + (z^9 - 2z^8 + z^7 + 2z^5 - z^4 + 3z^3 - 3z^2 + z - 2) * q^16 + (-3z^8 - 3z^6 - 2z^3 - z^2 + z + 2) q^17 + (2z^9 + z^7 - 3z^6 + 3z - 1) q^19 + (3z^9 - z^8 + z^7 - z^6 + z^5 - 3z^4 - 4z^2 + z - 4) q^20 + (z^9 + 2z^8 + 3z^7 + z^6 - z^3 - 3z^2 - 2z - 1) * q^22 + (-2z^9 + 4z^8 - 3z^7 + 3z^6 - 4z^5 + 2z^4 + 3z^2 - 3z + 3) * q^23 + (-z^9 - 2z^7 + 2z^5 - 2z^3 - z) q^25 + (2z^9 + 6z^7 + 2z^5 + 4z^3 - z^2 + z - 4) * q^26 + (-2z^9 + 3z^8 - 3z^7 - 3z^5 + 3z^4 - 2z^3 - 3z + 3) q^28 + (-3z^9 + 3z^8 - z^7 + 2z^6 - 2z^5 + z^4 - 3z^3 + 3z^2 - 1) * q^29 + (4z^9 - 3z^8 + 5z^7 - 2z^6 + 5z^5 - 3z^4 + 4z^3 + 4z - 4) * q^31 + (-2z^8 + z^6 + z^5 - z^4 - z^3 + 2z) * q^32 + (-3z^9 - 9z^7 + z^6 - 3z^5 + 3z^4 - 3z^3 + 3z^2 - z + 9) * q^34 + (-z^8 - z^7 + z^2 + z) * q^35 + (-2z^9 - 4z^8 + z^7 - z^6 + z^5 - z^4 + 4z^3 + 2z^2 - 3) * q^37 + (3z^9 - 4z^8 - z^7 - 2z^6 + 2z^5 + z^4 + 4z^3 - 3z^2 - 1) * q^38 + (2z^9 - 2z^8 - 3z^6 - 3z^4 - z^3 - 3z^2 - 3) q^40 + (-3z^7 + 3z^2 - 3z) q^41 + (-3z^9 + z^7 - 3z^5 - 2z^3 - 4z^2 + 4z + 2) q^43 + (2z^9 + 3z^8 - 3z^5 - 2z^4 - 5z^2 - z - 5) q^44 + (-5z^9 + 5z^8 - 3z^5 + 5z^4 + z^3 + 5z^2 - 3z) * q^46 + (-z^9 + 2z^8 + 3z^7 - 3z^6 - 2z^5 + z^4 + 2z^2 + 4z + 2) * q^47 + (-z^7 - 2z^6 + z^5 - 3z^4 + z^3 - 2z^2 - z) q^49 + (-z^8 - z^6 - 3z^3 - 3z^2 + 3z + 3) q^50 + (5z^9 + 7z^7 + 2z^6 - 2z - 7) * q^52 + (-5z^8 + 2z^7 - 4z^6 + 4z^5 - 5z^4 + 4z^3 - 4z^2 + 2z - 5) * q^53 + (-2z^9 - 5z^7 + 2z^6 - 5z^5 + 2z^4 - 5z^3 - 2z) q^55 + (-2z^9 - z^8 - 2z^7 - 2z^5 + 3z^4 + z^3 - z^2 - 3z + 2) q^56 + (2z^9 - 6z^8 + 6z^7 - 2z^6 - 4z^4 + 5z^3 - 2z^2 + 5z - 4) * q^58 + (-4z^9 + 7z^8 - z^7 + 4z^6 - z^5 + 7z^4 - 4z^3 - 5z + 5) * q^59 + (-3z^9 + 3z^8 + 2z^6 - 3z^5 - z^4 + 5z^3 - z^2 - 3z + 2) * q^61 + (9z^9 - 7z^8 + 9z^7 + 11z^5 - 6z^4 + 6z^3 - 6z^2 + 6z - 11) * q^62 + (-z^9 - 2z^8 - 6z^7 + 6z^6 + 2z^5 + z^4 + 2z^2 + 2z + 2) * q^64 + (4z^9 - 2z^8 + 2z^7 - 4z^6 - 2z^4 - 3z^3 - 2z^2 - 3z - 2) * q^65 + (-2z^9 - 2z^6 + 2z^5 - 4z^4 - 2z^3 + 4z^2 + 3z + 2) q^67 + (-6z^9 + z^8 - 7z^7 - z^6 + z^5 + 7z^4 - z^3 + 6z^2 + 14) * q^68 + (-z^9 - 2z^8 - z^7 + z^6 + 2z^5 + z^4 + z^2 + 2z + 1) q^70 + (-4z^8 + 5z^7 - 3z^6 + 6z^5 - 7z^4 + 6z^3 - 3z^2 + 5z - 4) * q^71 + (2z^9 - 2z^8 - 2z^6 + 2z^5 - 8z^3 + 2z - 2) * q^73 + (-9z^9 - 2z^8 + 2z^7 + 9z^6 - z^3 + 6z^2 - z) q^74 + (z^8 - z^7 + 3z^4 - 2z^3 - 2z + 3) q^76 + (-6z^9 + 4z^8 - 2z^7 + 5z^6 - 5z^5 + 5z^4 - 5z^3 + 2z^2 - 4z + 6) q^77 + (-4z^9 - 2z^8 + z^7 - 2z^6 + 3z^5 - 2z^4 + z^3 - 2z^2 - 4z) q^79 + (-2z^9 - 3z^7 + 2z^6 - 4z^5 + 2z^4 - 3z^3 - 2z) q^80 + (-6z^9 + 3z^4 - 3z^3) q^82 + (-z^9 + 4z^8 - z^7 - 2z^5 - 3z^4 - 3z^3 + 3z^2 + 3z + 2) * q^83 + (5z^7 + z^6 + 7z^5 - z^4 + 7z^3 + z^2 + 5z) * q^85 + (z^9 - 8z^7 + z^6 - 6z^5 - z^4 + z^3 + 6z^2 - z + 8) q^86 + (-z^9 + z^8 + z^6 - 5z^5 - 3z^4 - 5z^3 - 3z^2 - 5z + 1) q^88 + (7z^9 - z^8 + z^7 - z^6 + 3z^5 - 3z^4 + z^3 - z^2 + z - 7) q^89 + (-3z^9 - 2z^8 + 2z^7 - 2z^6 - 3z^5 + z^3 + z^2 - z - 1) q^91 + (-5z^9 + 6z^8 + z^7 + 6z^6 - 6z^5 + 6z^4 + z^3 + 6z^2 - 5z) q^92 + (3z^9 - 3z^8 + 3z^4 + 11z^3 + 3z^2) q^94 + (2z^9 - 2z^8 + 2z^7 - 3z^6 + 2z^5 - 2z^4 + 2z^3 + 2z - 2) * q^95 + (3z^9 + 3z^6 - 3z^5 - 3z^2 - 1) * q^97 + (-4z^9 + z^8 - 4z^7 - 2z^6 - 4z^5 + z^4 - 4z^3 - 2z + 2) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{2} + 10 q^{4} - 3 q^{5} - 6 q^{8}+O(q^{10}) \) 10 * q + 6 * q^2 + 10 * q^4 - 3 * q^5 - 6 * q^8	\( 10 q + 6 q^{2} + 10 q^{4} - 3 q^{5} - 6 q^{8} - 15 q^{10} - 15 q^{11} - 18 q^{13} - 6 q^{16} + 26 q^{17} - q^{19} - 25 q^{20} - 9 q^{22} + 6 q^{23} - 4 q^{25} - 24 q^{26} + 11 q^{28} - 28 q^{29} - 10 q^{31} + 4 q^{32} + 64 q^{34} - 22 q^{37} + 6 q^{38} - 18 q^{40} - 9 q^{41} + 21 q^{43} - 48 q^{44} - 25 q^{46} + 22 q^{47} + 7 q^{49} + 35 q^{50} - 62 q^{52} - 20 q^{53} - 23 q^{55} + 11 q^{56} - 8 q^{58} + 17 q^{59} + 13 q^{61} - 50 q^{62} + 10 q^{64} - 10 q^{65} + 23 q^{67} + 114 q^{68} + 11 q^{70} - q^{71} - 18 q^{73} - 22 q^{74} + 21 q^{76} + 22 q^{77} + 5 q^{79} - 18 q^{80} - 12 q^{82} + 12 q^{83} + 23 q^{85} + 61 q^{86} - 2 q^{88} - 51 q^{89} - 11 q^{91} - 38 q^{92} + 11 q^{94} - 3 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) \) 10 * q + 6 * q^2 + 10 * q^4 - 3 * q^5 - 6 * q^8 - 15 * q^10 - 15 * q^11 - 18 * q^13 - 6 * q^16 + 26 * q^17 - q^19 - 25 * q^20 - 9 * q^22 + 6 * q^23 - 4 * q^25 - 24 * q^26 + 11 * q^28 - 28 * q^29 - 10 * q^31 + 4 * q^32 + 64 * q^34 - 22 * q^37 + 6 * q^38 - 18 * q^40 - 9 * q^41 + 21 * q^43 - 48 * q^44 - 25 * q^46 + 22 * q^47 + 7 * q^49 + 35 * q^50 - 62 * q^52 - 20 * q^53 - 23 * q^55 + 11 * q^56 - 8 * q^58 + 17 * q^59 + 13 * q^61 - 50 * q^62 + 10 * q^64 - 10 * q^65 + 23 * q^67 + 114 * q^68 + 11 * q^70 - q^71 - 18 * q^73 - 22 * q^74 + 21 * q^76 + 22 * q^77 + 5 * q^79 - 18 * q^80 - 12 * q^82 + 12 * q^83 + 23 * q^85 + 61 * q^86 - 2 * q^88 - 51 * q^89 - 11 * q^91 - 38 * q^92 + 11 * q^94 - 3 * q^95 - 10 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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Label	603.2.u.b

Level	$603$
Weight	$2$
Character orbit	603.u
Analytic conductor	$4.815$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 67)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 64.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} - 6 T^{9} + \cdots + 1 \) T^10 - 6T^9 + 14T^8 - 18T^7 + 31T^6 - 54T^5 + 38T^4 - 52T^3 + 125T^2 + 9*T + 1
$3$	\( T^{10} \) T^10
$5$	\( T^{10} + 3 T^{9} + \cdots + 1 \) T^10 + 3T^9 + 9T^8 + 5T^7 + 15T^6 - 21T^5 + 14T^4 - 13T^3 + 16T^2 - 7*T + 1
$7$	\( T^{10} - 22 T^{7} + \cdots + 121 \) T^10 - 22T^7 - 33T^6 + 22T^5 + 363T^4 + 726T^3 + 726T^2 + 363*T + 121
$11$	\( T^{10} + 15 T^{9} + \cdots + 69169 \) T^10 + 15T^9 + 137T^8 + 845T^7 + 3765T^6 + 11661T^5 + 25821T^4 + 44313T^3 + 81541T^2 + 114931*T + 69169
$13$	\( T^{10} + 18 T^{9} + \cdots + 69169 \) T^10 + 18T^9 + 159T^8 + 1014T^7 + 5184T^6 + 19546T^5 + 50065T^4 + 75367T^3 + 56307T^2 + 23933*T + 69169
$17$	\( T^{10} - 26 T^{9} + \cdots + 737881 \) T^10 - 26T^9 + 390T^8 - 3837T^7 + 27283T^6 - 142473T^5 + 557154T^4 - 1557825T^3 + 2902513T^2 - 2430970*T + 737881
$19$	\( T^{10} + T^{9} + \cdots + 1849 \) T^10 + T^9 + 45T^8 + 45T^7 + 441T^6 - 3816T^5 + 3752T^4 - 4454T^3 + 31901T^2 - 12814T + 1849
$23$	\( T^{10} - 6 T^{9} + \cdots + 1067089 \) T^10 - 6T^9 + 69T^8 - 315T^7 + 1439T^6 - 2496T^5 - 4318T^4 - 10986T^3 + 236130T^2 - 816070*T + 1067089
$29$	\( (T^{5} + 14 T^{4} + \cdots + 199)^{2} \) (T^5 + 14T^4 + 41T^3 - 125T^2 - 483T + 199)^2
$31$	\( T^{10} + 10 T^{9} + \cdots + 2374681 \) T^10 + 10T^9 + 133T^8 + 505T^7 + 1618T^6 - 18536T^5 - 46925T^4 - 149051T^3 + 1214632T^2 + 2947933*T + 2374681
$37$	\( (T^{5} + 11 T^{4} + \cdots - 979)^{2} \) (T^5 + 11T^4 - 55T^3 - 451T^2 + 1804T - 979)^2
$41$	\( T^{10} + 9 T^{9} + \cdots + 59049 \) T^10 + 9T^9 + 81T^8 + 432T^7 + 324T^6 + 243T^5 + 18225T^4 - 4374T^3 + 32805T^2 + 78732*T + 59049
$43$	\( T^{10} - 21 T^{9} + \cdots + 83375161 \) T^10 - 21T^9 + 331T^8 - 3629T^7 + 23904T^6 - 132659T^5 + 922076T^4 - 6071735T^3 + 38549358T^2 - 89082036*T + 83375161
$47$	\( T^{10} - 22 T^{9} + \cdots + 33860761 \) T^10 - 22T^9 + 297T^8 - 2937T^7 + 31482T^6 - 273317T^5 + 1422234T^4 - 4945391T^3 + 16258286T^2 - 33860761*T + 33860761
$53$	\( T^{10} + 20 T^{9} + \cdots + 21132409 \) T^10 + 20T^9 + 169T^8 + 1048T^7 + 9410T^6 + 91730T^5 + 649229T^4 + 2988781T^3 + 8505269T^2 + 13427837*T + 21132409
$59$	\( T^{10} - 17 T^{9} + \cdots + 8826841 \) T^10 - 17T^9 + 300T^8 - 1360T^7 + 15970T^6 + 48665T^5 + 767673T^4 + 6113352T^3 + 22905920T^2 + 25093066*T + 8826841
$61$	\( T^{10} - 13 T^{9} + \cdots + 5139289 \) T^10 - 13T^9 + 235T^8 - 3594T^7 + 37383T^6 - 370655T^5 + 2561975T^4 - 7747945T^3 + 13119670T^2 - 12008299*T + 5139289
$67$	\( T^{10} + \cdots + 1350125107 \) T^10 - 23T^9 + 111T^8 + 967T^7 - 6379T^6 - 13685T^5 - 427393T^4 + 4340863T^3 + 33384693T^2 - 463475783*T + 1350125107
$71$	\( T^{10} + T^{9} + \cdots + 896809 \) T^10 + T^9 - 10T^8 + 1101T^7 + 18393T^6 + 18976T^5 + 209870T^4 + 262087T^3 + 898789T^2 + 1636416T + 896809
$73$	\( T^{10} + \cdots + 1786921984 \) T^10 + 18T^9 + 280T^8 + 4424T^7 + 49360T^6 + 464672T^5 + 3407936T^4 + 8718848T^3 - 18483456T^2 - 309092864*T + 1786921984
$79$	\( T^{10} - 5 T^{9} + \cdots + 4012009 \) T^10 - 5T^9 + 3T^8 - 1676T^7 + 5366T^6 + 55175T^5 + 770775T^4 + 3219136T^3 + 12377919T^2 + 11373034*T + 4012009
$83$	\( T^{10} - 12 T^{9} + \cdots + 20529961 \) T^10 - 12T^9 - 32T^8 + 1374T^7 + 7129T^6 + 21196T^5 + 143298T^4 + 166000T^3 + 2855183T^2 + 2895309*T + 20529961
$89$	\( T^{10} + \cdots + 375623161 \) T^10 + 51T^9 + 1215T^8 + 18207T^7 + 193075T^6 + 1515777T^5 + 9035679T^4 + 40680371T^3 + 135597163T^2 + 306200419*T + 375623161
$97$	\( (T^{5} + 5 T^{4} + \cdots - 989)^{2} \) (T^5 + 5T^4 - 89T^3 - 287T^2 + 1490T - 989)^2
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\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(\zeta_{22}^{4}\)	\(1\)