LMFDB - Newform orbit 3311.1.gd.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3311,1,Mod(146,3311)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3311, base_ring=CyclotomicField(210))

chi = DirichletCharacter(H, H._module([105, 168, 190]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3311.146");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3311 = 7 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3311.gd (of order $210$, degree $48$, 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:	$1.65240425683$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{210})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} - x^{47} + x^{46} + x^{43} - x^{42} + 2 x^{41} - x^{40} + x^{39} + x^{36} - x^{35} + x^{34} + \cdots + 1 $ x^48 - x^47 + x^46 + x^43 - x^42 + 2x^41 - x^40 + x^39 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 - x^28 - x^26 - x^24 - x^22 - x^20 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 + x^9 - x^8 + 2x^7 - x^6 + x^5 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{105}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{105} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times$.

$n$	$904$	$1893$	$2927$
$\chi(n)$	$-\zeta_{210}^{21}$	$-1$	$\zeta_{210}^{20}$

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} $

146.1

−0.791071	+	0.611724i
0.599822	−	0.800134i
−0.251587	−	0.967835i
−0.575617	+	0.817719i
0.946327	+	0.323210i
−0.420357	−	0.907359i
0.163818	−	0.986491i
0.842721	−	0.538351i
−0.971490	+	0.237080i
−0.998210	+	0.0598042i
−0.992847	−	0.119394i
−0.887586	+	0.460642i
−0.971490	−	0.237080i
0.280427	−	0.959875i
0.280427	+	0.959875i
−0.193256	−	0.981148i
−0.712376	−	0.701798i
−0.0149594	−	0.999888i
0.873408	+	0.486989i
0.447313	+	0.894377i

−0.673718

−

0.588609i

−0.0267988

−

0.197837i

0.669131

0.743145i

−0.591241

0.895691i

−0.337330

0.941386i

181.1

0.712420

−

1.07927i

−0.264256

−

0.618258i

−0.104528

−

0.994522i

0.416889

0.0756543i

0.0149594

0.999888i

412.1

−0.161032

1.18879i

−0.423320

−

0.116829i

−0.104528

−

0.994522i

−0.264437

0.618682i

−0.772417

0.635116i

454.1

0.0830272

−

1.84875i

−2.41499

−

0.217353i

−0.978148

0.207912i

−0.353927

2.61280i

−0.599822

0.800134i

482.1

0.984327

−

0.369424i

0.0793539

−

0.0693294i

0.913545

−

0.406737i

−0.445712

0.828271i

0.575617

−

0.817719i

531.1

−0.0709825

1.58055i

−1.49712

−

0.134743i

0.669131

0.743145i

0.106862

−

0.788887i

0.992847

0.119394i

685.1

−1.71874

0.311906i

1.92056

−

0.720799i

−0.978148

0.207912i

−1.57658

0.941963i

0.887586

−

0.460642i

713.1

1.30794

−

1.36800i

−0.115847

−

2.57954i

0.913545

−

0.406737i

−2.25503

−

1.97016i

0.998210

−

0.0598042i

741.1

0.607158

−

1.42052i

−0.958167

−

1.00216i

0.669131

−

0.743145i

−0.559023

0.209805i

0.525684

−

0.850680i

762.1

0.545534

−

1.01377i

−0.179229

−

0.271520i

−0.978148

0.207912i

0.773565

−

0.0696221i

0.251587

0.967835i

797.1

0.371668

−

0.563053i

0.214133

0.500990i

0.913545

0.406737i

1.02549

0.186099i

−0.873408

−

0.486989i

874.1

−0.267104

−

0.279369i

0.0381624

−

0.849752i

−0.104528

−

0.994522i

−0.538659

0.470612i

−0.447313

−

0.894377i

916.1

0.607158

1.42052i

−0.958167

1.00216i

0.669131

0.743145i

−0.559023

−

0.209805i

0.525684

0.850680i

1049.1

0.128769

−

0.301270i

0.616880

0.645206i

−0.978148

−

0.207912i

0.580561

−

0.217888i

−0.999552

−

0.0299155i

1070.1

0.128769

0.301270i

0.616880

−

0.645206i

−0.978148

0.207912i

0.580561

0.217888i

−0.999552

0.0299155i

1098.1

−1.87163

0.702435i

2.25652

−

1.97146i

−0.104528

0.994522i

−1.89124

3.51450i

0.420357

0.907359i

1175.1

0.266546

1.96772i

−2.83692

0.782941i

0.913545

−

0.406737i

−1.51635

−

3.54769i

−0.163818

0.986491i

1357.1

−1.87296

0.516904i

2.38234

−

1.42338i

0.669131

−

0.743145i

−2.38356

2.49301i

−0.946327

−

0.323210i

1434.1

0.540643

−

0.149208i

−0.588417

0.351563i

−0.978148

−

0.207912i

−0.653253

0.683249i

0.193256

0.981148i

1588.1

0.398388

0.740329i

0.161523

−

0.244696i

0.669131

−

0.743145i

1.08283

0.0974569i

0.712376

0.701798i

See all 48 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
473.bc	even	105	1	inner
3311.gd	odd	210	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3311.1.gd.a	✓	48
7.b	odd	2	1	CM	3311.1.gd.a	✓	48
11.c	even	5	1		3311.1.gd.b	yes	48
43.g	even	21	1		3311.1.gd.b	yes	48
77.j	odd	10	1		3311.1.gd.b	yes	48
301.bc	odd	42	1		3311.1.gd.b	yes	48
473.bc	even	105	1	inner	3311.1.gd.a	✓	48
3311.gd	odd	210	1	inner	3311.1.gd.a	✓	48

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3311.1.gd.a	✓	48	1.a	even	1	1	trivial
3311.1.gd.a	✓	48	7.b	odd	2	1	CM
3311.1.gd.a	✓	48	473.bc	even	105	1	inner
3311.1.gd.a	✓	48	3311.gd	odd	210	1	inner
3311.1.gd.b	yes	48	11.c	even	5	1
3311.1.gd.b	yes	48	43.g	even	21	1
3311.1.gd.b	yes	48	77.j	odd	10	1
3311.1.gd.b	yes	48	301.bc	odd	42	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{48} + 2 T_{2}^{47} + T_{2}^{46} + 23 T_{2}^{45} + 41 T_{2}^{44} + 25 T_{2}^{43} + 341 T_{2}^{42} + \cdots + 1 $ T2^48 + 2*T2^47 + T2^46 + 23*T2^45 + 41*T2^44 + 25*T2^43 + 341*T2^42 + 514*T2^41 + 270*T2^40 + 3178*T2^39 + 4092*T2^38 + 2120*T2^37 + 22299*T2^36 + 23407*T2^35 + 10433*T2^34 + 115220*T2^33 + 96350*T2^32 + 38522*T2^31 + 457074*T2^30 + 259351*T2^29 + 63086*T2^28 + 1319308*T2^27 + 225358*T2^26 - 175094*T2^25 + 2640128*T2^24 - 782957*T2^23 - 300439*T2^22 + 3642149*T2^21 - 2529149*T2^20 + 1249738*T2^19 + 3436804*T2^18 - 2368171*T2^17 + 1728932*T2^16 + 2557559*T2^15 - 205581*T2^14 - 1508803*T2^13 + 2835029*T2^12 - 855618*T2^11 - 624485*T2^10 + 610028*T2^9 - 108388*T2^8 - 97058*T2^7 + 23338*T2^6 + 15474*T2^5 + 1206*T2^4 - 366*T2^3 + 1134*T2^2 - 66*T2 + 1 acting on $S_{1}^{\mathrm{new}}(3311, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 + T^46 + 23T^45 + 41T^44 + 25T^43 + 341T^42 + 514T^41 + 270T^40 + 3178T^39 + 4092T^38 + 2120T^37 + 22299T^36 + 23407T^35 + 10433T^34 + 115220T^33 + 96350T^32 + 38522T^31 + 457074T^30 + 259351T^29 + 63086T^28 + 1319308T^27 + 225358T^26 - 175094T^25 + 2640128T^24 - 782957T^23 - 300439T^22 + 3642149T^21 - 2529149T^20 + 1249738T^19 + 3436804T^18 - 2368171T^17 + 1728932T^16 + 2557559T^15 - 205581T^14 - 1508803T^13 + 2835029T^12 - 855618T^11 - 624485T^10 + 610028T^9 - 108388T^8 - 97058T^7 + 23338T^6 + 15474T^5 + 1206T^4 - 366T^3 + 1134T^2 - 66T + 1
$3$	$ T^{48} $ T^48
$5$	$ T^{48} $ T^48
$7$	$ (T^{8} - T^{7} + T^{5} + \cdots + 1)^{6} $ (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^6
$11$	$ (T^{24} - T^{23} + T^{19} + \cdots + 1)^{2} $ (T^24 - T^23 + T^19 - T^18 + T^17 - T^16 + T^14 - T^13 + T^12 - T^11 + T^10 - T^8 + T^7 - T^6 + T^5 - T + 1)^2
$13$	$ T^{48} $ T^48
$17$	$ T^{48} $ T^48
$19$	$ T^{48} $ T^48
$23$	$ T^{48} - T^{47} + \cdots + 1 $ T^48 - T^47 - 4T^46 + 5T^45 - 7T^44 + 31T^43 + 47T^42 - 188T^41 - 33T^40 - 115T^39 + 604T^38 + 2574T^37 - 2299T^36 - 6423T^35 - 3064T^34 + 5049T^33 + 32963T^32 + 30167T^31 - 72982T^30 - 169409T^29 + 79397T^28 + 156154T^27 + 272399T^26 + 248299T^25 - 826726T^24 - 384319T^23 + 951963T^22 + 294791T^21 + 771527T^20 + 476474T^19 + 839197T^18 + 649766T^17 - 3183699T^16 - 2149697T^15 + 3477519T^14 + 2338670T^13 - 1334479T^12 - 15928T^11 + 1044317T^10 + 237986T^9 + 134656T^8 + 137795T^7 + 28979T^6 + 18629T^5 + 20964T^4 + 7842T^3 + 955T^2 - 17T + 1
$29$	$ T^{48} - T^{47} + \cdots + 1 $ T^48 - T^47 + T^46 - 5T^45 + 14T^44 - 27T^43 + 15T^42 + 119T^40 - 176T^39 - 324T^38 + 774T^37 - 326T^36 - 1245T^35 - 2684T^34 + 14179T^33 + 3305T^32 - 49534T^31 + 12720T^30 + 123881T^29 - 105324T^28 - 251980T^27 + 574039T^26 - 327202T^25 + 107433T^24 - 1828229T^23 + 6446452T^22 - 12328820T^21 + 18436618T^20 - 29441324T^19 + 54153624T^18 - 95171170T^17 + 141262527T^16 - 172283646T^15 + 174310078T^14 - 148241328T^13 + 107448247T^12 - 67005320T^11 + 36250811T^10 - 17094947T^9 + 7049116T^8 - 2552260T^7 + 802627T^6 - 217369T^5 + 54380T^4 - 12193T^3 + 1671T^2 - 68T + 1
$31$	$ T^{48} $ T^48
$37$	$ T^{48} - 3 T^{47} + \cdots + 1 $ T^48 - 3T^47 - 3T^46 + 16T^45 - 4T^44 - 19T^43 + 45T^42 - 145T^41 - 119T^40 + 917T^39 + 470T^38 - 4060T^37 + 1300T^36 + 6087T^35 - 14216T^34 + 36424T^33 + 20377T^32 - 205031T^31 + 111401T^30 + 299935T^29 - 101880T^28 - 173644T^27 - 635881T^26 - 164653T^25 + 1468227T^24 + 3264784T^23 - 3139893T^22 - 9567027T^21 + 8490901T^20 + 1827553T^19 + 7821692T^18 - 10107577T^17 + 1630253T^16 - 10493873T^15 + 7828300T^14 + 5888234T^13 - 1801515T^12 - 1442570T^11 + 571265T^10 - 3737175T^9 + 1712610T^8 + 952817T^7 - 53212T^6 + 539T^5 + 16234T^4 + 2272T^3 + 207T^2 + 19*T + 1
$41$	$ T^{48} $ T^48
$43$	$ T^{48} + T^{47} + \cdots + 1 $ T^48 + T^47 + T^46 - T^43 - T^42 - 2T^41 - T^40 - T^39 + T^36 + T^35 + T^34 + T^33 + T^32 + T^31 - T^28 - T^26 - T^24 - T^22 - T^20 + T^17 + T^16 + T^15 + T^14 + T^13 + T^12 - T^9 - T^8 - 2T^7 - T^6 - T^5 + T^2 + T + 1
$47$	$ T^{48} $ T^48
$53$	$ T^{48} - 10 T^{47} + \cdots + 1 $ T^48 - 10T^47 + 46T^46 - 124T^45 + 199T^44 - 124T^43 - 284T^42 + 1101T^41 - 2205T^40 + 3262T^39 - 4003T^38 + 4928T^37 - 9711T^36 + 36145T^35 - 131151T^34 + 336276T^33 - 568876T^32 + 587012T^31 - 222842T^30 - 589170T^29 + 2703622T^28 - 8391378T^27 + 19594420T^26 - 35928192T^25 + 57202486T^24 - 87504678T^23 + 131530146T^22 - 187277000T^21 + 247394069T^20 - 302740690T^19 + 343102057T^18 - 361760011T^17 + 355033198T^16 - 321308594T^15 + 268853813T^14 - 209479156T^13 + 147939594T^12 - 90863937T^11 + 49233613T^10 - 22965832T^9 + 7717847T^8 - 1638100T^7 + 386451T^6 - 78302T^5 + 1891T^4 + 242T^3 + 179T^2 + 12*T + 1
$59$	$ T^{48} $ T^48
$61$	$ T^{48} $ T^48
$67$	$ T^{48} - T^{47} + \cdots + 1 $ T^48 - T^47 - 4T^46 - 2T^45 + 7T^44 + 17T^43 + 33T^42 - 27T^41 - 89T^40 - 108T^39 - 348T^38 - 44T^37 + 2328T^36 + 4007T^35 + 2529T^34 - 15566T^33 - 40698T^32 - 2166T^31 + 91511T^30 + 71587T^29 + 32245T^28 + 57573T^27 - 126174T^26 - 49873T^25 + 213327T^24 - 188130T^23 + 163637T^22 + 19187T^21 - 196937T^20 + 36230T^19 + 404875T^18 - 917576T^17 + 450862T^16 + 1707842T^15 - 3737388T^14 + 2273367T^13 + 4966753T^12 - 16568975T^11 + 26136664T^10 - 27100605T^9 + 19795458T^8 - 10417085T^7 + 3959038T^6 - 1070459T^5 + 197973T^4 - 23014T^3 + 1382T^2 - 24T + 1
$71$	$ T^{48} + 9 T^{47} + \cdots + 1 $ T^48 + 9T^47 + 46T^46 + 175T^45 + 550T^44 + 1506T^43 + 3704T^42 + 8427T^41 + 18164T^40 + 37666T^39 + 75620T^38 + 146810T^37 + 273776T^36 + 484784T^35 + 811886T^34 + 1293689T^33 + 1988411T^32 + 2993794T^31 + 4435335T^30 + 6287360T^29 + 7654314T^28 + 5910980T^27 - 3722091T^26 - 27172300T^25 - 68054046T^24 - 121667240T^23 - 169402176T^22 - 184194400T^21 - 140087971T^20 - 21234195T^19 + 173956285T^18 + 427771134T^17 + 681056506T^16 + 803586569T^15 + 705700266T^14 + 426192239T^13 + 131274816T^12 - 12885365T^11 - 6290700T^10 + 303866T^9 + 306724T^8 + 16482T^7 - 2631T^6 + 2516T^5 + 225T^4 - 100T^3 - 14T^2 + 4*T + 1
$73$	$ T^{48} $ T^48
$79$	$ T^{48} - 6 T^{47} + \cdots + 1 $ T^48 - 6T^47 + 9T^46 + 22T^45 - 103T^44 + 122T^43 + 201T^42 - 1033T^41 + 1459T^40 + 1127T^39 - 7141T^38 + 8351T^37 + 6016T^36 - 31938T^35 + 42601T^34 + 10030T^33 - 120209T^32 + 93196T^31 + 239360T^30 - 486179T^29 - 12426T^28 + 978401T^27 - 975709T^26 - 1228735T^25 + 4588428T^24 - 4689854T^23 - 1048683T^22 + 7151739T^21 - 6788690T^20 + 835324T^19 + 4964729T^18 - 4511974T^17 - 575737T^16 + 1035595T^15 + 4166203T^14 - 6067414T^13 + 2728458T^12 + 257701T^11 - 465748T^10 + 27108T^9 + 205035T^8 - 340141T^7 + 254240T^6 - 110092T^5 + 38773T^4 - 10712T^3 + 1518T^2 - 53*T + 1
$83$	$ T^{48} $ T^48
$89$	$ T^{48} $ T^48
$97$	$ T^{48} $ T^48
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:	\( N \)	\(=\)	\( 3311 = 7 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3311.gd (of order \(210\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{210}^{34} + \zeta_{210}^{2}) q^{2} + (\zeta_{210}^{68} + \cdots + \zeta_{210}^{4}) q^{4}+ \cdots - \zeta_{210}^{83} q^{9} +O(q^{10}) \) q + (z^34 + z^2) * q^2 + (z^68 + z^36 + z^4) * q^4 + z^56 * q^7 + (z^102 + z^70 + z^38 + z^6) * q^8 - z^83 * q^9	\( q + (\zeta_{210}^{34} + \zeta_{210}^{2}) q^{2} + (\zeta_{210}^{68} + \cdots + \zeta_{210}^{4}) q^{4}+ \cdots + \zeta_{210}^{50} q^{99}+O(q^{100}) \) q + (z^34 + z^2) * q^2 + (z^68 + z^36 + z^4) * q^4 + z^56 * q^7 + (z^102 + z^70 + z^38 + z^6) * q^8 - z^83 * q^9 + z^72 * q^11 + (z^90 + z^58) * q^14 + (z^104 + z^72 + z^40 - z^31 + z^8) * q^16 + (-z^85 + z^12) * q^18 + (z^74 - z) * q^22 + (-z^99 - z^11) * q^23 - z^59 * q^25 + (z^92 + z^60 - z^19) * q^28 + (-z^73 + z^54) * q^29 + (z^74 - z^65 + z^42 + z^33 + z^10 + z) * q^32 + (-z^87 + z^46 + z^14) * q^36 + (z^100 + z^82) * q^37 + z^92 * q^43 + (z^76 - z^35 - z^3) * q^44 + (-z^101 - z^45 + z^28 - z^13) * q^46 - z^7 * q^49 + (-z^93 - z^61) * q^50 + (-z^91 - z^25) * q^53 + (z^94 + z^62 - z^53 - z^21) * q^56 + (z^88 - z^75 + z^56 + z^2) * q^58 + z^34 * q^63 + (-z^99 + z^76 - z^67 + z^44 + z^35 + z^12 + z^3) * q^64 + (-z^47 + z^18) * q^67 + (z^64 - z^15) * q^71 + (-z^89 + z^80 + z^48 + z^16) * q^72 + (z^102 + z^84 - z^29 - z^11) * q^74 - z^23 * q^77 + (-z^85 + z^6) * q^79 - z^61 * q^81 + (z^94 - z^21) * q^86 + (z^78 - z^69 - z^37 - z^5) * q^88 + (-z^103 - z^79 + z^62 - z^47 + z^30 - z^15) * q^92 + (-z^41 - z^9) * q^98 + z^50 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{2} + 6 q^{7} - 21 q^{8} - q^{9}+O(q^{10}) \) 48 * q - 2 * q^2 + 6 * q^7 - 21 * q^8 - q^9	\( 48 q - 2 q^{2} + 6 q^{7} - 21 q^{8} - q^{9} + 2 q^{11} - 9 q^{14} + 3 q^{16} + 6 q^{18} - 2 q^{22} + q^{23} - q^{25} - 10 q^{28} + q^{29} - 4 q^{32} + 7 q^{36} + 3 q^{37} - q^{43} - 23 q^{44} - 4 q^{46} + 6 q^{49} + q^{50} + 10 q^{53} - 15 q^{56} - 4 q^{58} - q^{63} - 21 q^{64} + q^{67} - 9 q^{71} + 4 q^{72} - 12 q^{74} - q^{77} + 6 q^{79} - q^{81} - 13 q^{86} + 7 q^{88} - 20 q^{92} + q^{98} + 4 q^{99}+O(q^{100}) \) 48 * q - 2 * q^2 + 6 * q^7 - 21 * q^8 - q^9 + 2 * q^11 - 9 * q^14 + 3 * q^16 + 6 * q^18 - 2 * q^22 + q^23 - q^25 - 10 * q^28 + q^29 - 4 * q^32 + 7 * q^36 + 3 * q^37 - q^43 - 23 * q^44 - 4 * q^46 + 6 * q^49 + q^50 + 10 * q^53 - 15 * q^56 - 4 * q^58 - q^63 - 21 * q^64 + q^67 - 9 * q^71 + 4 * q^72 - 12 * q^74 - q^77 + 6 * q^79 - q^81 - 13 * q^86 + 7 * q^88 - 20 * q^92 + 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	3311.1.gd.a

Level	$3311$
Weight	$1$
Character orbit	3311.gd
Analytic conductor	$1.652$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{105}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.65240425683\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Coefficient field:	\(\Q(\zeta_{210})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{48} - x^{47} + x^{46} + x^{43} - x^{42} + 2 x^{41} - x^{40} + x^{39} + x^{36} - x^{35} + x^{34} + \cdots + 1 \) x^48 - x^47 + x^46 + x^43 - x^42 + 2x^41 - x^40 + x^39 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 - x^28 - x^26 - x^24 - x^22 - x^20 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 + x^9 - x^8 + 2x^7 - x^6 + x^5 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{105}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{105} - \cdots)\)

\(n\)	\(904\)	\(1893\)	\(2927\)
\(\chi(n)\)	\(-\zeta_{210}^{21}\)	\(-1\)	\(\zeta_{210}^{20}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 146.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
473.bc	even	105	1	inner
3311.gd	odd	210	1	inner

$p$	$F_p(T)$
$2$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 + T^46 + 23T^45 + 41T^44 + 25T^43 + 341T^42 + 514T^41 + 270T^40 + 3178T^39 + 4092T^38 + 2120T^37 + 22299T^36 + 23407T^35 + 10433T^34 + 115220T^33 + 96350T^32 + 38522T^31 + 457074T^30 + 259351T^29 + 63086T^28 + 1319308T^27 + 225358T^26 - 175094T^25 + 2640128T^24 - 782957T^23 - 300439T^22 + 3642149T^21 - 2529149T^20 + 1249738T^19 + 3436804T^18 - 2368171T^17 + 1728932T^16 + 2557559T^15 - 205581T^14 - 1508803T^13 + 2835029T^12 - 855618T^11 - 624485T^10 + 610028T^9 - 108388T^8 - 97058T^7 + 23338T^6 + 15474T^5 + 1206T^4 - 366T^3 + 1134T^2 - 66T + 1
$3$	\( T^{48} \) T^48
$5$	\( T^{48} \) T^48
$7$	\( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{6} \) (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^6
$11$	\( (T^{24} - T^{23} + T^{19} + \cdots + 1)^{2} \) (T^24 - T^23 + T^19 - T^18 + T^17 - T^16 + T^14 - T^13 + T^12 - T^11 + T^10 - T^8 + T^7 - T^6 + T^5 - T + 1)^2
$13$	\( T^{48} \) T^48
$17$	\( T^{48} \) T^48
$19$	\( T^{48} \) T^48
$23$	\( T^{48} - T^{47} + \cdots + 1 \) T^48 - T^47 - 4T^46 + 5T^45 - 7T^44 + 31T^43 + 47T^42 - 188T^41 - 33T^40 - 115T^39 + 604T^38 + 2574T^37 - 2299T^36 - 6423T^35 - 3064T^34 + 5049T^33 + 32963T^32 + 30167T^31 - 72982T^30 - 169409T^29 + 79397T^28 + 156154T^27 + 272399T^26 + 248299T^25 - 826726T^24 - 384319T^23 + 951963T^22 + 294791T^21 + 771527T^20 + 476474T^19 + 839197T^18 + 649766T^17 - 3183699T^16 - 2149697T^15 + 3477519T^14 + 2338670T^13 - 1334479T^12 - 15928T^11 + 1044317T^10 + 237986T^9 + 134656T^8 + 137795T^7 + 28979T^6 + 18629T^5 + 20964T^4 + 7842T^3 + 955T^2 - 17T + 1
$29$	\( T^{48} - T^{47} + \cdots + 1 \) T^48 - T^47 + T^46 - 5T^45 + 14T^44 - 27T^43 + 15T^42 + 119T^40 - 176T^39 - 324T^38 + 774T^37 - 326T^36 - 1245T^35 - 2684T^34 + 14179T^33 + 3305T^32 - 49534T^31 + 12720T^30 + 123881T^29 - 105324T^28 - 251980T^27 + 574039T^26 - 327202T^25 + 107433T^24 - 1828229T^23 + 6446452T^22 - 12328820T^21 + 18436618T^20 - 29441324T^19 + 54153624T^18 - 95171170T^17 + 141262527T^16 - 172283646T^15 + 174310078T^14 - 148241328T^13 + 107448247T^12 - 67005320T^11 + 36250811T^10 - 17094947T^9 + 7049116T^8 - 2552260T^7 + 802627T^6 - 217369T^5 + 54380T^4 - 12193T^3 + 1671T^2 - 68T + 1
$31$	\( T^{48} \) T^48
$37$	\( T^{48} - 3 T^{47} + \cdots + 1 \) T^48 - 3T^47 - 3T^46 + 16T^45 - 4T^44 - 19T^43 + 45T^42 - 145T^41 - 119T^40 + 917T^39 + 470T^38 - 4060T^37 + 1300T^36 + 6087T^35 - 14216T^34 + 36424T^33 + 20377T^32 - 205031T^31 + 111401T^30 + 299935T^29 - 101880T^28 - 173644T^27 - 635881T^26 - 164653T^25 + 1468227T^24 + 3264784T^23 - 3139893T^22 - 9567027T^21 + 8490901T^20 + 1827553T^19 + 7821692T^18 - 10107577T^17 + 1630253T^16 - 10493873T^15 + 7828300T^14 + 5888234T^13 - 1801515T^12 - 1442570T^11 + 571265T^10 - 3737175T^9 + 1712610T^8 + 952817T^7 - 53212T^6 + 539T^5 + 16234T^4 + 2272T^3 + 207T^2 + 19*T + 1
$41$	\( T^{48} \) T^48
$43$	\( T^{48} + T^{47} + \cdots + 1 \) T^48 + T^47 + T^46 - T^43 - T^42 - 2T^41 - T^40 - T^39 + T^36 + T^35 + T^34 + T^33 + T^32 + T^31 - T^28 - T^26 - T^24 - T^22 - T^20 + T^17 + T^16 + T^15 + T^14 + T^13 + T^12 - T^9 - T^8 - 2T^7 - T^6 - T^5 + T^2 + T + 1
$47$	\( T^{48} \) T^48
$53$	\( T^{48} - 10 T^{47} + \cdots + 1 \) T^48 - 10T^47 + 46T^46 - 124T^45 + 199T^44 - 124T^43 - 284T^42 + 1101T^41 - 2205T^40 + 3262T^39 - 4003T^38 + 4928T^37 - 9711T^36 + 36145T^35 - 131151T^34 + 336276T^33 - 568876T^32 + 587012T^31 - 222842T^30 - 589170T^29 + 2703622T^28 - 8391378T^27 + 19594420T^26 - 35928192T^25 + 57202486T^24 - 87504678T^23 + 131530146T^22 - 187277000T^21 + 247394069T^20 - 302740690T^19 + 343102057T^18 - 361760011T^17 + 355033198T^16 - 321308594T^15 + 268853813T^14 - 209479156T^13 + 147939594T^12 - 90863937T^11 + 49233613T^10 - 22965832T^9 + 7717847T^8 - 1638100T^7 + 386451T^6 - 78302T^5 + 1891T^4 + 242T^3 + 179T^2 + 12*T + 1
$59$	\( T^{48} \) T^48
$61$	\( T^{48} \) T^48
$67$	\( T^{48} - T^{47} + \cdots + 1 \) T^48 - T^47 - 4T^46 - 2T^45 + 7T^44 + 17T^43 + 33T^42 - 27T^41 - 89T^40 - 108T^39 - 348T^38 - 44T^37 + 2328T^36 + 4007T^35 + 2529T^34 - 15566T^33 - 40698T^32 - 2166T^31 + 91511T^30 + 71587T^29 + 32245T^28 + 57573T^27 - 126174T^26 - 49873T^25 + 213327T^24 - 188130T^23 + 163637T^22 + 19187T^21 - 196937T^20 + 36230T^19 + 404875T^18 - 917576T^17 + 450862T^16 + 1707842T^15 - 3737388T^14 + 2273367T^13 + 4966753T^12 - 16568975T^11 + 26136664T^10 - 27100605T^9 + 19795458T^8 - 10417085T^7 + 3959038T^6 - 1070459T^5 + 197973T^4 - 23014T^3 + 1382T^2 - 24T + 1
$71$	\( T^{48} + 9 T^{47} + \cdots + 1 \) T^48 + 9T^47 + 46T^46 + 175T^45 + 550T^44 + 1506T^43 + 3704T^42 + 8427T^41 + 18164T^40 + 37666T^39 + 75620T^38 + 146810T^37 + 273776T^36 + 484784T^35 + 811886T^34 + 1293689T^33 + 1988411T^32 + 2993794T^31 + 4435335T^30 + 6287360T^29 + 7654314T^28 + 5910980T^27 - 3722091T^26 - 27172300T^25 - 68054046T^24 - 121667240T^23 - 169402176T^22 - 184194400T^21 - 140087971T^20 - 21234195T^19 + 173956285T^18 + 427771134T^17 + 681056506T^16 + 803586569T^15 + 705700266T^14 + 426192239T^13 + 131274816T^12 - 12885365T^11 - 6290700T^10 + 303866T^9 + 306724T^8 + 16482T^7 - 2631T^6 + 2516T^5 + 225T^4 - 100T^3 - 14T^2 + 4*T + 1
$73$	\( T^{48} \) T^48
$79$	\( T^{48} - 6 T^{47} + \cdots + 1 \) T^48 - 6T^47 + 9T^46 + 22T^45 - 103T^44 + 122T^43 + 201T^42 - 1033T^41 + 1459T^40 + 1127T^39 - 7141T^38 + 8351T^37 + 6016T^36 - 31938T^35 + 42601T^34 + 10030T^33 - 120209T^32 + 93196T^31 + 239360T^30 - 486179T^29 - 12426T^28 + 978401T^27 - 975709T^26 - 1228735T^25 + 4588428T^24 - 4689854T^23 - 1048683T^22 + 7151739T^21 - 6788690T^20 + 835324T^19 + 4964729T^18 - 4511974T^17 - 575737T^16 + 1035595T^15 + 4166203T^14 - 6067414T^13 + 2728458T^12 + 257701T^11 - 465748T^10 + 27108T^9 + 205035T^8 - 340141T^7 + 254240T^6 - 110092T^5 + 38773T^4 - 10712T^3 + 1518T^2 - 53*T + 1
$83$	\( T^{48} \) T^48
$89$	\( T^{48} \) T^48
$97$	\( T^{48} \) T^48
show more
show less