LMFDB - Newform orbit 69.2.e.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,2,Mod(4,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("69.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	69.e (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:	$0.550967773947$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-\zeta_{22}$	$1$

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} $

4.1

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

0.915415

2.00448i

0.959493

−

0.281733i

−1.87023

2.15836i

−2.61903

−

1.68315i

1.44306

1.66538i

−0.427961

−

2.97653i

−1.80972

−

0.531382i

0.841254

−

0.540641i

0.976337

−

6.79057i

13.1

−0.154861

−

0.178719i

−0.841254

−

0.540641i

0.276671

−

1.92429i

0.418382

−

0.916128i

0.0336545

0.234072i

1.97204

0.579043i

−0.784630

0.504251i

0.415415

0.909632i

−0.228520

0.0670996i

16.1

−0.154861

0.178719i

−0.841254

0.540641i

0.276671

1.92429i

0.418382

0.916128i

0.0336545

−

0.234072i

1.97204

−

0.579043i

−0.784630

−

0.504251i

0.415415

−

0.909632i

−0.228520

−

0.0670996i

25.1

1.34125

0.861971i

0.142315

0.989821i

0.225136

0.492980i

−2.43560

−

0.715158i

−0.662317

1.45027i

0.729022

−

0.841336i

0.330830

−

2.30097i

−0.959493

0.281733i

−2.65032

−

3.05863i

31.1

0.357685

−

2.48775i

−0.415415

−

0.909632i

−4.14200

−

1.21620i

2.65565

3.06479i

−2.41153

0.708089i

−1.15843

−

0.744479i

−2.41899

5.29684i

−0.654861

0.755750i

8.57432

−

5.51038i

49.1

0.357685

2.48775i

−0.415415

0.909632i

−4.14200

1.21620i

2.65565

−

3.06479i

−2.41153

−

0.708089i

−1.15843

0.744479i

−2.41899

−

5.29684i

−0.654861

−

0.755750i

8.57432

5.51038i

52.1

0.915415

−

2.00448i

0.959493

0.281733i

−1.87023

−

2.15836i

−2.61903

1.68315i

1.44306

−

1.66538i

−0.427961

2.97653i

−1.80972

0.531382i

0.841254

0.540641i

0.976337

6.79057i

55.1

−0.459493

0.134919i

0.654861

0.755750i

−1.48958

0.957293i

0.480602

3.34266i

−0.402869

−

0.258908i

1.88533

−

4.12830i

1.18251

−

1.36469i

−0.142315

0.989821i

−0.671822

−

1.47109i

58.1

1.34125

−

0.861971i

0.142315

−

0.989821i

0.225136

−

0.492980i

−2.43560

0.715158i

−0.662317

−

1.45027i

0.729022

0.841336i

0.330830

2.30097i

−0.959493

−

0.281733i

−2.65032

3.05863i

64.1

−0.459493

−

0.134919i

0.654861

−

0.755750i

−1.48958

−

0.957293i

0.480602

−

3.34266i

−0.402869

0.258908i

1.88533

4.12830i

1.18251

1.36469i

−0.142315

−

0.989821i

−0.671822

1.47109i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	69.2.e.b	✓	10
3.b	odd	2	1		207.2.i.a		10
23.c	even	11	1	inner	69.2.e.b	✓	10
23.c	even	11	1		1587.2.a.q		5
23.d	odd	22	1		1587.2.a.r		5
69.g	even	22	1		4761.2.a.bm		5
69.h	odd	22	1		207.2.i.a		10
69.h	odd	22	1		4761.2.a.bp		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
69.2.e.b	✓	10	1.a	even	1	1	trivial
69.2.e.b	✓	10	23.c	even	11	1	inner
207.2.i.a		10	3.b	odd	2	1
207.2.i.a		10	69.h	odd	22	1
1587.2.a.q		5	23.c	even	11	1
1587.2.a.r		5	23.d	odd	22	1
4761.2.a.bm		5	69.g	even	22	1
4761.2.a.bp		5	69.h	odd	22	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} - 4T_{2}^{9} + 16T_{2}^{8} - 31T_{2}^{7} + 47T_{2}^{6} - 23T_{2}^{5} - 18T_{2}^{4} + 39T_{2}^{3} + 31T_{2}^{2} + 8T_{2} + 1 $ T2^10 - 4*T2^9 + 16*T2^8 - 31*T2^7 + 47*T2^6 - 23*T2^5 - 18*T2^4 + 39*T2^3 + 31*T2^2 + 8*T2 + 1 acting on $S_{2}^{\mathrm{new}}(69, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - 4 T^{9} + \cdots + 1 $ T^10 - 4T^9 + 16T^8 - 31T^7 + 47T^6 - 23T^5 - 18T^4 + 39T^3 + 31T^2 + 8*T + 1
$3$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	$ T^{10} + 3 T^{9} + \cdots + 11881 $ T^10 + 3T^9 + 9T^8 + 71T^7 + 290T^6 + 782T^5 + 3204T^4 + 8336T^3 + 6737T^2 + 763*T + 11881
$7$	$ T^{10} - 6 T^{9} + \cdots + 1849 $ T^10 - 6T^9 + 36T^8 - 95T^7 + 240T^6 - 472T^5 + 93T^4 + 597T^3 + 576T^2 - 1806*T + 1849
$11$	$ T^{10} + 15 T^{9} + \cdots + 1 $ T^10 + 15T^9 + 137T^8 + 746T^7 + 2676T^6 + 5754T^5 + 7209T^4 + 4856T^3 + 1384T^2 + 14*T + 1
$13$	$ T^{10} - 8 T^{9} + \cdots + 1 $ T^10 - 8T^9 + 31T^8 - 39T^7 - 18T^6 + 23T^5 + 47T^4 + 31T^3 + 16T^2 + 4*T + 1
$17$	$ T^{10} - T^{9} + \cdots + 39601 $ T^10 - T^9 + T^8 - 111T^7 + 111T^6 + 615T^5 + 2190T^4 - 2465T^3 + 24740T^2 - 35223*T + 39601
$19$	$ T^{10} + 9 T^{9} + \cdots + 109561 $ T^10 + 9T^9 + 48T^8 + 124T^7 - 28T^6 + 529T^5 + 3573T^4 - 7718T^3 + 42056T^2 - 96652*T + 109561
$23$	$ T^{10} - 21 T^{9} + \cdots + 6436343 $ T^10 - 21T^9 + 221T^8 - 1627T^7 + 9505T^6 - 47937T^5 + 218615T^4 - 860683T^3 + 2688907T^2 - 5876661*T + 6436343
$29$	$ T^{10} + 8 T^{9} + \cdots + 109561 $ T^10 + 8T^9 + 108T^8 + 743T^7 + 4272T^6 + 27103T^5 + 115052T^4 + 270569T^3 + 375303T^2 + 293597*T + 109561
$31$	$ T^{10} + 23 T^{9} + \cdots + 11881 $ T^10 + 23T^9 + 298T^8 + 2487T^7 + 14697T^6 + 52383T^5 + 95085T^4 - 99824T^3 + 12035T^2 - 12099*T + 11881
$37$	$ T^{10} - 3 T^{9} + \cdots + 4190209 $ T^10 - 3T^9 - 57T^8 - 137T^7 + 2919T^6 - 3268T^5 + 88102T^4 - 144516T^3 + 1033785T^2 - 2034718*T + 4190209
$41$	$ T^{10} + \cdots + 203889841 $ T^10 + 15T^9 + 137T^8 + 372T^7 + 2896T^6 + 4038T^5 + 13094T^4 - 1042685T^3 + 3215309T^2 - 34826481*T + 203889841
$43$	$ T^{10} - 22 T^{9} + \cdots + 1437601 $ T^10 - 22T^9 + 220T^8 - 759T^7 - 660T^6 - 6633T^5 + 55176T^4 + 254947T^3 + 735317T^2 + 1147443*T + 1437601
$47$	$ (T^{5} - 2 T^{4} + \cdots - 13133)^{2} $ (T^5 - 2T^4 - 148T^3 + 365T^2 + 5064T - 13133)^2
$53$	$ T^{10} - 29 T^{9} + \cdots + 55965361 $ T^10 - 29T^9 + 434T^8 - 5634T^7 + 73296T^6 - 747889T^5 + 4803407T^4 - 16284428T^3 + 43238776T^2 - 69678034*T + 55965361
$59$	$ T^{10} + \cdots + 474411961 $ T^10 + 54T^9 + 1431T^8 + 23847T^7 + 270832T^6 + 2150587T^5 + 11903068T^4 + 45244121T^3 + 133825286T^2 + 314321611*T + 474411961
$61$	$ T^{10} + 30 T^{9} + \cdots + 591361 $ T^10 + 30T^9 + 504T^8 + 5440T^7 + 46193T^6 + 186856T^5 + 921198T^4 + 2908798T^3 + 15157235T^2 - 5842093*T + 591361
$67$	$ T^{10} - T^{9} + \cdots + 6285049 $ T^10 - T^9 - 32T^8 + 131T^7 + 5479T^6 - 103357T^5 + 496189T^4 - 1085206T^3 + 11220727T^2 + 10757537T + 6285049
$71$	$ T^{10} + 3 T^{9} + \cdots + 21743569 $ T^10 + 3T^9 + 163T^8 - 941T^7 + 10443T^6 - 41381T^5 + 40406T^4 + 748405T^3 - 1903248T^2 - 1916493*T + 21743569
$73$	$ T^{10} + 47 T^{9} + \cdots + 58081 $ T^10 + 47T^9 + 1197T^8 + 18892T^7 + 201700T^6 + 1422433T^5 + 6224023T^4 + 15303594T^3 + 16654951T^2 - 228950*T + 58081
$79$	$ T^{10} - 18 T^{9} + \cdots + 34774609 $ T^10 - 18T^9 + 16T^8 - 783T^7 + 21189T^6 + 100464T^5 + 965980T^4 + 1218145T^3 + 7610920T^2 - 29237326*T + 34774609
$83$	$ T^{10} + \cdots + 141681409 $ T^10 - 18T^9 + 335T^8 - 5535T^7 + 79346T^6 - 810501T^5 + 6243582T^4 - 33988191T^3 + 127482474T^2 - 222026659*T + 141681409
$89$	$ T^{10} - 25 T^{9} + \cdots + 69169 $ T^10 - 25T^9 + 438T^8 - 4009T^7 + 23621T^6 - 73646T^5 + 151704T^4 - 189693T^3 + 110555T^2 + 238278*T + 69169
$97$	$ T^{10} - 21 T^{9} + \cdots + 39601 $ T^10 - 21T^9 + 210T^8 - 1440T^7 + 6755T^6 - 17940T^5 + 43055T^4 + 57432T^3 + 359844T^2 - 203378*T + 39601
show more
show less

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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

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	69.2.e.b

Level	$69$
Weight	$2$
Character orbit	69.e
Analytic conductor	$0.551$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.550967773947\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 4.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} - 4 T^{9} + \cdots + 1 \) T^10 - 4T^9 + 16T^8 - 31T^7 + 47T^6 - 23T^5 - 18T^4 + 39T^3 + 31T^2 + 8*T + 1
$3$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	\( T^{10} + 3 T^{9} + \cdots + 11881 \) T^10 + 3T^9 + 9T^8 + 71T^7 + 290T^6 + 782T^5 + 3204T^4 + 8336T^3 + 6737T^2 + 763*T + 11881
$7$	\( T^{10} - 6 T^{9} + \cdots + 1849 \) T^10 - 6T^9 + 36T^8 - 95T^7 + 240T^6 - 472T^5 + 93T^4 + 597T^3 + 576T^2 - 1806*T + 1849
$11$	\( T^{10} + 15 T^{9} + \cdots + 1 \) T^10 + 15T^9 + 137T^8 + 746T^7 + 2676T^6 + 5754T^5 + 7209T^4 + 4856T^3 + 1384T^2 + 14*T + 1
$13$	\( T^{10} - 8 T^{9} + \cdots + 1 \) T^10 - 8T^9 + 31T^8 - 39T^7 - 18T^6 + 23T^5 + 47T^4 + 31T^3 + 16T^2 + 4*T + 1
$17$	\( T^{10} - T^{9} + \cdots + 39601 \) T^10 - T^9 + T^8 - 111T^7 + 111T^6 + 615T^5 + 2190T^4 - 2465T^3 + 24740T^2 - 35223*T + 39601
$19$	\( T^{10} + 9 T^{9} + \cdots + 109561 \) T^10 + 9T^9 + 48T^8 + 124T^7 - 28T^6 + 529T^5 + 3573T^4 - 7718T^3 + 42056T^2 - 96652*T + 109561
$23$	\( T^{10} - 21 T^{9} + \cdots + 6436343 \) T^10 - 21T^9 + 221T^8 - 1627T^7 + 9505T^6 - 47937T^5 + 218615T^4 - 860683T^3 + 2688907T^2 - 5876661*T + 6436343
$29$	\( T^{10} + 8 T^{9} + \cdots + 109561 \) T^10 + 8T^9 + 108T^8 + 743T^7 + 4272T^6 + 27103T^5 + 115052T^4 + 270569T^3 + 375303T^2 + 293597*T + 109561
$31$	\( T^{10} + 23 T^{9} + \cdots + 11881 \) T^10 + 23T^9 + 298T^8 + 2487T^7 + 14697T^6 + 52383T^5 + 95085T^4 - 99824T^3 + 12035T^2 - 12099*T + 11881
$37$	\( T^{10} - 3 T^{9} + \cdots + 4190209 \) T^10 - 3T^9 - 57T^8 - 137T^7 + 2919T^6 - 3268T^5 + 88102T^4 - 144516T^3 + 1033785T^2 - 2034718*T + 4190209
$41$	\( T^{10} + \cdots + 203889841 \) T^10 + 15T^9 + 137T^8 + 372T^7 + 2896T^6 + 4038T^5 + 13094T^4 - 1042685T^3 + 3215309T^2 - 34826481*T + 203889841
$43$	\( T^{10} - 22 T^{9} + \cdots + 1437601 \) T^10 - 22T^9 + 220T^8 - 759T^7 - 660T^6 - 6633T^5 + 55176T^4 + 254947T^3 + 735317T^2 + 1147443*T + 1437601
$47$	\( (T^{5} - 2 T^{4} + \cdots - 13133)^{2} \) (T^5 - 2T^4 - 148T^3 + 365T^2 + 5064T - 13133)^2
$53$	\( T^{10} - 29 T^{9} + \cdots + 55965361 \) T^10 - 29T^9 + 434T^8 - 5634T^7 + 73296T^6 - 747889T^5 + 4803407T^4 - 16284428T^3 + 43238776T^2 - 69678034*T + 55965361
$59$	\( T^{10} + \cdots + 474411961 \) T^10 + 54T^9 + 1431T^8 + 23847T^7 + 270832T^6 + 2150587T^5 + 11903068T^4 + 45244121T^3 + 133825286T^2 + 314321611*T + 474411961
$61$	\( T^{10} + 30 T^{9} + \cdots + 591361 \) T^10 + 30T^9 + 504T^8 + 5440T^7 + 46193T^6 + 186856T^5 + 921198T^4 + 2908798T^3 + 15157235T^2 - 5842093*T + 591361
$67$	\( T^{10} - T^{9} + \cdots + 6285049 \) T^10 - T^9 - 32T^8 + 131T^7 + 5479T^6 - 103357T^5 + 496189T^4 - 1085206T^3 + 11220727T^2 + 10757537T + 6285049
$71$	\( T^{10} + 3 T^{9} + \cdots + 21743569 \) T^10 + 3T^9 + 163T^8 - 941T^7 + 10443T^6 - 41381T^5 + 40406T^4 + 748405T^3 - 1903248T^2 - 1916493*T + 21743569
$73$	\( T^{10} + 47 T^{9} + \cdots + 58081 \) T^10 + 47T^9 + 1197T^8 + 18892T^7 + 201700T^6 + 1422433T^5 + 6224023T^4 + 15303594T^3 + 16654951T^2 - 228950*T + 58081
$79$	\( T^{10} - 18 T^{9} + \cdots + 34774609 \) T^10 - 18T^9 + 16T^8 - 783T^7 + 21189T^6 + 100464T^5 + 965980T^4 + 1218145T^3 + 7610920T^2 - 29237326*T + 34774609
$83$	\( T^{10} + \cdots + 141681409 \) T^10 - 18T^9 + 335T^8 - 5535T^7 + 79346T^6 - 810501T^5 + 6243582T^4 - 33988191T^3 + 127482474T^2 - 222026659*T + 141681409
$89$	\( T^{10} - 25 T^{9} + \cdots + 69169 \) T^10 - 25T^9 + 438T^8 - 4009T^7 + 23621T^6 - 73646T^5 + 151704T^4 - 189693T^3 + 110555T^2 + 238278*T + 69169
$97$	\( T^{10} - 21 T^{9} + \cdots + 39601 \) T^10 - 21T^9 + 210T^8 - 1440T^7 + 6755T^6 - 17940T^5 + 43055T^4 + 57432T^3 + 359844T^2 - 203378*T + 39601
show more
show less

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(-\zeta_{22}\)	\(1\)