LMFDB - Newform orbit 3311.1.gb.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3311,1,Mod(62,3311)]

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, 147, 95]))

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

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

chi := DirichletCharacter("3311.62");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3311 = 7 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3311.gb (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:	$D_{210}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{210} - \cdots)$

Character values

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

$n$	$904$	$1893$	$2927$
$\chi(n)$	$-\zeta_{210}^{84}$	$-1$	$-\zeta_{210}^{10}$

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} $

62.1

−0.791071	+	0.611724i
0.946327	−	0.323210i
0.646600	+	0.762830i
−0.998210	−	0.0598042i
−0.998210	+	0.0598042i
0.447313	+	0.894377i
0.599822	+	0.800134i
−0.525684	+	0.850680i
0.873408	+	0.486989i
0.447313	−	0.894377i
−0.992847	−	0.119394i
−0.575617	−	0.817719i
−0.971490	+	0.237080i
0.337330	−	0.941386i
0.280427	−	0.959875i
−0.193256	−	0.981148i
0.999552	+	0.0299155i
0.772417	−	0.635116i
−0.575617	+	0.817719i
−0.971490	−	0.237080i

−1.78392

0.492330i

2.08153

−

1.24365i

−0.669131

0.743145i

−1.82210

1.90576i

0.946327

0.323210i

244.1

−0.0518827

−

0.383014i

0.819955

−

0.226293i

−0.913545

0.406737i

−0.281124

−

0.657722i

0.163818

−

0.986491i

321.1

0.631640

−

0.237059i

−0.410299

0.358467i

0.104528

−

0.994522i

−0.493884

0.917790i

−0.420357

−

0.907359i

349.1

−0.351611

−

0.822634i

0.137966

−

0.144302i

0.978148

−

0.207912i

−1.00480

−

0.377107i

0.999552

−

0.0299155i

370.1

−0.351611

0.822634i

0.137966

0.144302i

0.978148

0.207912i

−1.00480

0.377107i

0.999552

0.0299155i

503.1

0.784643

1.83576i

−2.06330

2.15804i

−0.669131

−

0.743145i

−3.71150

−

1.39295i

−0.525684

−

0.850680i

545.1

1.37224

1.43525i

−0.132036

2.94002i

0.104528

0.994522i

−2.90547

2.53844i

0.447313

0.894377i

622.1

−1.10130

1.66840i

−1.17767

−

2.75529i

−0.913545

−

0.406737i

3.92692

0.712630i

0.873408

0.486989i

657.1

−0.0141776

0.0263464i

0.550404

0.833826i

0.978148

−

0.207912i

−0.0595701

0.00536140i

−0.251587

−

0.967835i

678.1

0.784643

−

1.83576i

−2.06330

−

2.15804i

−0.669131

0.743145i

−3.71150

1.39295i

−0.525684

0.850680i

706.1

−0.829029

0.867096i

−0.0197022

−

0.438703i

−0.913545

0.406737i

−0.506686

−

0.442678i

−0.998210

0.0598042i

734.1

−0.827204

0.150115i

−0.274503

0.103023i

0.978148

−

0.207912i

0.933315

−

0.557630i

−0.887586

0.460642i

888.1

−0.0251627

0.560290i

0.682683

0.0614425i

−0.669131

−

0.743145i

−0.126889

0.936733i

−0.992847

−

0.119394i

937.1

1.21074

−

0.454398i

0.506338

−

0.442374i

−0.913545

0.406737i

−0.200777

0.373107i

−0.575617

0.817719i

965.1

0.0871715

−

1.94102i

−2.76400

−

0.248764i

0.978148

−

0.207912i

−0.462987

3.41791i

0.599822

−

0.800134i

1007.1

0.254057

−

1.87553i

−2.48909

−

0.686945i

0.104528

0.994522i

−1.17689

2.75348i

0.772417

−

0.635116i

1238.1

0.579195

−

0.877443i

−0.0414145

−

0.0968940i

0.104528

0.994522i

0.925466

0.167947i

−0.0149594

−

0.999888i

1273.1

0.246733

0.215564i

−0.119824

−

0.884576i

−0.669131

−

0.743145i

0.341612

−

0.517520i

0.337330

−

0.941386i

1448.1

−0.827204

−

0.150115i

−0.274503

−

0.103023i

0.978148

0.207912i

0.933315

0.557630i

−0.887586

−

0.460642i

1525.1

−0.0251627

−

0.560290i

0.682683

−

0.0614425i

−0.669131

0.743145i

−0.126889

−

0.936733i

−0.992847

0.119394i

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.bf	even	210	1	inner
3311.gb	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.gb.b	yes	48
7.b	odd	2	1	CM	3311.1.gb.b	yes	48
11.d	odd	10	1		3311.1.gb.a	✓	48
43.h	odd	42	1		3311.1.gb.a	✓	48
77.l	even	10	1		3311.1.gb.a	✓	48
301.bn	even	42	1		3311.1.gb.a	✓	48
473.bf	even	210	1	inner	3311.1.gb.b	yes	48
3311.gb	odd	210	1	inner	3311.1.gb.b	yes	48

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3311.1.gb.a	✓	48	11.d	odd	10	1
3311.1.gb.a	✓	48	43.h	odd	42	1
3311.1.gb.a	✓	48	77.l	even	10	1
3311.1.gb.a	✓	48	301.bn	even	42	1
3311.1.gb.b	yes	48	1.a	even	1	1	trivial
3311.1.gb.b	yes	48	7.b	odd	2	1	CM
3311.1.gb.b	yes	48	473.bf	even	210	1	inner
3311.1.gb.b	yes	48	3311.gb	odd	210	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{48} - 2 T_{2}^{47} + T_{2}^{46} + 12 T_{2}^{45} - 29 T_{2}^{44} + 10 T_{2}^{43} + 61 T_{2}^{42} + \cdots + 1 $ T2^48 - 2*T2^47 + T2^46 + 12*T2^45 - 29*T2^44 + 10*T2^43 + 61*T2^42 - 129*T2^41 - 80*T2^40 + 112*T2^39 + 767*T2^38 - 2365*T2^37 - 346*T2^36 + 14113*T2^35 - 20157*T2^34 - 14630*T2^33 + 88125*T2^32 - 80872*T2^31 - 144506*T2^30 + 368304*T2^29 - 56544*T2^28 - 796933*T2^27 + 822003*T2^26 + 997489*T2^25 - 2073042*T2^24 - 817733*T2^23 + 3781086*T2^22 + 895881*T2^21 - 4495904*T2^20 - 767683*T2^19 + 3383499*T2^18 + 1034496*T2^17 - 1793783*T2^16 - 1200084*T2^15 + 2939414*T2^14 + 2852138*T2^13 - 464771*T2^12 - 698102*T2^11 - 292790*T2^10 - 126783*T2^9 + 225617*T2^8 + 135418*T2^7 + 29743*T2^6 + 4091*T2^5 + 3796*T2^4 - 649*T2^3 + 1099*T2^2 + 31*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 + 12T^45 - 29T^44 + 10T^43 + 61T^42 - 129T^41 - 80T^40 + 112T^39 + 767T^38 - 2365T^37 - 346T^36 + 14113T^35 - 20157T^34 - 14630T^33 + 88125T^32 - 80872T^31 - 144506T^30 + 368304T^29 - 56544T^28 - 796933T^27 + 822003T^26 + 997489T^25 - 2073042T^24 - 817733T^23 + 3781086T^22 + 895881T^21 - 4495904T^20 - 767683T^19 + 3383499T^18 + 1034496T^17 - 1793783T^16 - 1200084T^15 + 2939414T^14 + 2852138T^13 - 464771T^12 - 698102T^11 - 292790T^10 - 126783T^9 + 225617T^8 + 135418T^7 + 29743T^6 + 4091T^5 + 3796T^4 - 649T^3 + 1099T^2 + 31T + 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 - 41T^44 + 28T^43 - 40T^42 + 200T^41 + 744T^40 - 86T^39 + 546T^38 - 2826T^37 - 8841T^36 - 3855T^35 - 5754T^34 + 18649T^33 + 71325T^32 + 55846T^31 + 71315T^30 - 67194T^29 - 314554T^28 - 405360T^27 - 418616T^26 - 159487T^25 + 642743T^24 + 1059701T^23 + 1668767T^22 + 757190T^21 + 231863T^20 + 338456T^19 + 1073899T^18 + 612010T^17 + 3164287T^16 + 5635239T^15 + 6240158T^14 + 2803647T^13 + 965707T^12 + 689790T^11 + 1069431T^10 + 317158T^9 + 68916T^8 - 16460T^7 + 80087T^6 + 11306T^5 + 320T^4 - 448T^3 + 1136T^2 - 48*T + 1
$31$	$ T^{48} $ T^48
$37$	$ T^{48} - 5 T^{47} + \cdots + 1 $ T^48 - 5T^47 + 17T^46 - 40T^45 + 64T^44 - 55T^43 - 113T^42 + 625T^41 - 1775T^40 + 3475T^39 - 5152T^38 + 5030T^37 + 708T^36 - 15825T^35 + 52156T^34 - 111080T^33 + 205249T^32 - 340045T^31 + 473911T^30 - 556125T^29 + 348854T^28 + 286650T^27 - 1639641T^26 + 3465015T^25 - 5161195T^24 + 5478680T^23 - 2692483T^22 - 1925125T^21 + 8054059T^20 - 10611275T^19 + 11833184T^18 - 9774535T^17 + 8556367T^16 - 6199955T^15 + 4913300T^14 - 3058130T^13 + 1926779T^12 - 1263140T^11 + 459827T^10 - 235635T^9 + 122688T^8 - 17945T^7 + 7412T^6 - 3775T^5 - 114T^4 - 30T^3 + 37T^2 + 5*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} + 25 T^{47} + \cdots + 1 $ T^48 + 25T^47 + 326T^46 + 2921T^45 + 20079T^44 + 112331T^43 + 530526T^42 + 2168371T^41 + 7807730T^40 + 25101167T^39 + 72802857T^38 + 192074953T^37 + 464048639T^36 + 1032319810T^35 + 2124271164T^34 + 4058999521T^33 + 7225228024T^32 + 12014467142T^31 + 18706614853T^30 + 27326638625T^29 + 37514792957T^28 + 48466532077T^27 + 58990045505T^26 + 67695860618T^25 + 73284077536T^24 + 74850835022T^23 + 72117769576T^22 + 65507483635T^21 + 56038598139T^20 + 45075313290T^19 + 34013375022T^18 + 23999500689T^17 + 15760667078T^16 + 9571062881T^15 + 5328967683T^14 + 2688092829T^13 + 1202421239T^12 + 455715103T^11 + 133867603T^10 + 26444648T^9 + 3609722T^8 + 1338090T^7 + 700646T^6 + 165648T^5 + 11481T^4 - 1088T^3 - 66T^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} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 - 2T^46 - 10T^45 - 5T^44 + 28T^43 + 61T^42 + 18T^41 - 202T^40 - 448T^39 + 106T^38 + 2057T^37 + 2790T^36 - 3781T^35 - 2990T^34 + 15517T^33 + 18390T^32 - 15514T^31 - 50977T^30 - 70731T^29 + 27943T^28 + 8292T^27 - 234916T^26 - 172584T^25 + 242892T^24 + 256468T^23 + 350311T^22 + 344174T^21 + 346818T^20 + 419422T^19 + 372317T^18 + 200092T^17 + 165795T^16 + 102081T^15 + 66800T^14 + 38478T^13 + 3300T^12 - 7484T^11 + 4934T^10 - 1136T^9 + 268T^8 + 309T^7 - 66T^6 + 56T^5 - 20T^4 + 5T^3 + 2T^2 - T + 1
$73$	$ T^{48} $ T^48
$79$	$ T^{48} + 7 T^{47} + \cdots + 1 $ T^48 + 7T^47 + 35T^46 + 134T^45 + 430T^44 + 1211T^43 + 3025T^42 + 6877T^41 + 14239T^40 + 27175T^39 + 47330T^38 + 74165T^37 + 100227T^36 + 101496T^35 + 31954T^34 - 193016T^33 - 655982T^32 - 1397482T^31 - 2278553T^30 - 2845425T^29 - 2352223T^28 + 193581T^27 + 5068149T^26 + 11734269T^25 + 17459669T^24 + 19029812T^23 + 13532816T^22 + 226322T^21 - 14226737T^20 - 25945928T^19 - 25249537T^18 - 19522330T^17 - 12259865T^16 - 10500077T^15 - 9278920T^14 - 1308887T^13 + 12891827T^12 + 24901480T^11 + 26685146T^10 + 19726617T^9 + 10794198T^8 + 4737856T^7 + 1699844T^6 + 421343T^5 + 61236T^4 + 4824T^3 + 226T^2 + 11*T + 1
$83$	$ T^{48} $ T^48
$89$	$ T^{48} $ T^48
$97$	$ T^{48} $ T^48
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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:	\( N \)	\(=\)	\( 3311 = 7 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3311.gb (of order \(210\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{210}^{38} + \zeta_{210}) q^{2} + (\zeta_{210}^{76} + \cdots + \zeta_{210}^{2}) q^{4}+ \cdots - \zeta_{210}^{52} q^{9} +O(q^{10}) \) q + (-z^38 + z) * q^2 + (z^76 - z^39 + z^2) * q^4 + z^49 * q^7 + (z^77 - z^40 + z^9 + z^3) * q^8 - z^52 * q^9	\( q + ( - \zeta_{210}^{38} + \zeta_{210}) q^{2} + (\zeta_{210}^{76} + \cdots + \zeta_{210}^{2}) q^{4}+ \cdots - \zeta_{210}^{85} q^{99} +O(q^{100}) \) q + (-z^38 + z) * q^2 + (z^76 - z^39 + z^2) * q^4 + z^49 * q^7 + (z^77 - z^40 + z^9 + z^3) * q^8 - z^52 * q^9 + z^33 * q^11 + (-z^87 + z^50) * q^14 + (z^78 - z^47 - z^41 - z^10 + z^4) * q^16 + (z^90 - z^53) * q^18 + (-z^71 + z^34) * q^22 + (z^39 - z^16) * q^23 + z^61 * q^25 + (-z^88 + z^51 - z^20) * q^28 + (-z^27 + z^26) * q^29 + (z^85 + z^79 - z^48 - z^42 - z^11 + z^5) * q^32 + (z^91 - z^54 + z^23) * q^36 + (z^20 - z^8) * q^37 - z^67 * q^43 + (-z^72 + z^35 - z^4) * q^44 + (-z^77 + z^54 + z^40 - z^17) * q^46 + z^98 * q^49 + (-z^99 + z^62) * q^50 + (z^44 - z^35) * q^53 + (-z^89 + z^58 + z^52 - z^21) * q^56 + (z^65 - z^64 - z^28 + z^27) * q^58 - z^101 * q^63 + (z^86 + z^80 - z^49 - z^43 + z^18 - z^12 + z^6) * q^64 + (-z^97 - z^93) * q^67 + (z^95 - z^81) * q^71 + (z^92 - z^61 - z^55 + z^24) * q^72 + (-z^58 + z^46 + z^21 - z^9) * q^74 + z^82 * q^77 + (-z^74 - z^45) * q^79 + z^104 * q^81 + (-z^68 - 1) * q^86 + (-z^73 + z^42 + z^36 - z^5) * q^88 + (-z^92 - z^78 + z^55 + z^41 - z^18 - z^10) * q^92 + (z^99 + z^31) * q^98 - z^85 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{2} - 6 q^{7} - 14 q^{8} + q^{9}+O(q^{10}) \) 48 * q + 2 * q^2 - 6 * q^7 - 14 * q^8 + q^9	\( 48 q + 2 q^{2} - 6 q^{7} - 14 q^{8} + q^{9} - 2 q^{11} + 6 q^{14} + 3 q^{16} - 9 q^{18} - 2 q^{22} - q^{23} + q^{25} - 5 q^{28} + q^{29} + 4 q^{32} - 7 q^{36} + 5 q^{37} - q^{43} + 23 q^{44} + 11 q^{46} + 6 q^{49} + q^{50} - 25 q^{53} - 15 q^{56} - 11 q^{58} - q^{63} + 14 q^{64} + q^{67} - 2 q^{71} + 4 q^{72} + 14 q^{74} - q^{77} - 7 q^{79} - q^{81} - 47 q^{86} - 7 q^{88} - 10 q^{92} - q^{98} + 4 q^{99}+O(q^{100}) \) 48 * q + 2 * q^2 - 6 * q^7 - 14 * q^8 + q^9 - 2 * q^11 + 6 * q^14 + 3 * q^16 - 9 * q^18 - 2 * q^22 - q^23 + q^25 - 5 * q^28 + q^29 + 4 * q^32 - 7 * q^36 + 5 * q^37 - q^43 + 23 * q^44 + 11 * q^46 + 6 * q^49 + q^50 - 25 * q^53 - 15 * q^56 - 11 * q^58 - q^63 + 14 * q^64 + q^67 - 2 * q^71 + 4 * q^72 + 14 * q^74 - q^77 - 7 * q^79 - q^81 - 47 * q^86 - 7 * q^88 - 10 * q^92 - q^98 + 4 * q^99

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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	3311.1.gb.b

Level	$3311$
Weight	$1$
Character orbit	3311.gb
Analytic conductor	$1.652$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{210}$
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:	\(D_{210}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{210} - \cdots)\)

\(n\)	\(904\)	\(1893\)	\(2927\)
\(\chi(n)\)	\(-\zeta_{210}^{84}\)	\(-1\)	\(-\zeta_{210}^{10}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 62.1
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.bf	even	210	1	inner
3311.gb	odd	210	1	inner

$p$	$F_p(T)$
$2$	\( T^{48} - 2 T^{47} + \cdots + 1 \) T^48 - 2T^47 + T^46 + 12T^45 - 29T^44 + 10T^43 + 61T^42 - 129T^41 - 80T^40 + 112T^39 + 767T^38 - 2365T^37 - 346T^36 + 14113T^35 - 20157T^34 - 14630T^33 + 88125T^32 - 80872T^31 - 144506T^30 + 368304T^29 - 56544T^28 - 796933T^27 + 822003T^26 + 997489T^25 - 2073042T^24 - 817733T^23 + 3781086T^22 + 895881T^21 - 4495904T^20 - 767683T^19 + 3383499T^18 + 1034496T^17 - 1793783T^16 - 1200084T^15 + 2939414T^14 + 2852138T^13 - 464771T^12 - 698102T^11 - 292790T^10 - 126783T^9 + 225617T^8 + 135418T^7 + 29743T^6 + 4091T^5 + 3796T^4 - 649T^3 + 1099T^2 + 31T + 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 - 41T^44 + 28T^43 - 40T^42 + 200T^41 + 744T^40 - 86T^39 + 546T^38 - 2826T^37 - 8841T^36 - 3855T^35 - 5754T^34 + 18649T^33 + 71325T^32 + 55846T^31 + 71315T^30 - 67194T^29 - 314554T^28 - 405360T^27 - 418616T^26 - 159487T^25 + 642743T^24 + 1059701T^23 + 1668767T^22 + 757190T^21 + 231863T^20 + 338456T^19 + 1073899T^18 + 612010T^17 + 3164287T^16 + 5635239T^15 + 6240158T^14 + 2803647T^13 + 965707T^12 + 689790T^11 + 1069431T^10 + 317158T^9 + 68916T^8 - 16460T^7 + 80087T^6 + 11306T^5 + 320T^4 - 448T^3 + 1136T^2 - 48*T + 1
$31$	\( T^{48} \) T^48
$37$	\( T^{48} - 5 T^{47} + \cdots + 1 \) T^48 - 5T^47 + 17T^46 - 40T^45 + 64T^44 - 55T^43 - 113T^42 + 625T^41 - 1775T^40 + 3475T^39 - 5152T^38 + 5030T^37 + 708T^36 - 15825T^35 + 52156T^34 - 111080T^33 + 205249T^32 - 340045T^31 + 473911T^30 - 556125T^29 + 348854T^28 + 286650T^27 - 1639641T^26 + 3465015T^25 - 5161195T^24 + 5478680T^23 - 2692483T^22 - 1925125T^21 + 8054059T^20 - 10611275T^19 + 11833184T^18 - 9774535T^17 + 8556367T^16 - 6199955T^15 + 4913300T^14 - 3058130T^13 + 1926779T^12 - 1263140T^11 + 459827T^10 - 235635T^9 + 122688T^8 - 17945T^7 + 7412T^6 - 3775T^5 - 114T^4 - 30T^3 + 37T^2 + 5*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
$53$	\( T^{48} + 25 T^{47} + \cdots + 1 \) T^48 + 25T^47 + 326T^46 + 2921T^45 + 20079T^44 + 112331T^43 + 530526T^42 + 2168371T^41 + 7807730T^40 + 25101167T^39 + 72802857T^38 + 192074953T^37 + 464048639T^36 + 1032319810T^35 + 2124271164T^34 + 4058999521T^33 + 7225228024T^32 + 12014467142T^31 + 18706614853T^30 + 27326638625T^29 + 37514792957T^28 + 48466532077T^27 + 58990045505T^26 + 67695860618T^25 + 73284077536T^24 + 74850835022T^23 + 72117769576T^22 + 65507483635T^21 + 56038598139T^20 + 45075313290T^19 + 34013375022T^18 + 23999500689T^17 + 15760667078T^16 + 9571062881T^15 + 5328967683T^14 + 2688092829T^13 + 1202421239T^12 + 455715103T^11 + 133867603T^10 + 26444648T^9 + 3609722T^8 + 1338090T^7 + 700646T^6 + 165648T^5 + 11481T^4 - 1088T^3 - 66T^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
$71$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 - 2T^46 - 10T^45 - 5T^44 + 28T^43 + 61T^42 + 18T^41 - 202T^40 - 448T^39 + 106T^38 + 2057T^37 + 2790T^36 - 3781T^35 - 2990T^34 + 15517T^33 + 18390T^32 - 15514T^31 - 50977T^30 - 70731T^29 + 27943T^28 + 8292T^27 - 234916T^26 - 172584T^25 + 242892T^24 + 256468T^23 + 350311T^22 + 344174T^21 + 346818T^20 + 419422T^19 + 372317T^18 + 200092T^17 + 165795T^16 + 102081T^15 + 66800T^14 + 38478T^13 + 3300T^12 - 7484T^11 + 4934T^10 - 1136T^9 + 268T^8 + 309T^7 - 66T^6 + 56T^5 - 20T^4 + 5T^3 + 2T^2 - T + 1
$73$	\( T^{48} \) T^48
$79$	\( T^{48} + 7 T^{47} + \cdots + 1 \) T^48 + 7T^47 + 35T^46 + 134T^45 + 430T^44 + 1211T^43 + 3025T^42 + 6877T^41 + 14239T^40 + 27175T^39 + 47330T^38 + 74165T^37 + 100227T^36 + 101496T^35 + 31954T^34 - 193016T^33 - 655982T^32 - 1397482T^31 - 2278553T^30 - 2845425T^29 - 2352223T^28 + 193581T^27 + 5068149T^26 + 11734269T^25 + 17459669T^24 + 19029812T^23 + 13532816T^22 + 226322T^21 - 14226737T^20 - 25945928T^19 - 25249537T^18 - 19522330T^17 - 12259865T^16 - 10500077T^15 - 9278920T^14 - 1308887T^13 + 12891827T^12 + 24901480T^11 + 26685146T^10 + 19726617T^9 + 10794198T^8 + 4737856T^7 + 1699844T^6 + 421343T^5 + 61236T^4 + 4824T^3 + 226T^2 + 11*T + 1
$83$	\( T^{48} \) T^48
$89$	\( T^{48} \) T^48
$97$	\( T^{48} \) T^48
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