LMFDB - Newform orbit 3784.1.em.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3784,1,Mod(203,3784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3784, base_ring=CyclotomicField(210))

chi = DirichletCharacter(H, H._module([105, 105, 84, 170]))

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

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

chi := DirichletCharacter("3784.203");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3784 = 2^{3} \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3784.em (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:	$1.88846200780$
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:	$\Z[a_1, a_2, a_3]$
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 $\left(\mathbb{Z}/3784\mathbb{Z}\right)^\times$.

$n$	$89$	$1377$	$1893$	$2839$
$\chi(n)$	$\zeta_{210}^{40}$	$-\zeta_{210}^{63}$	$-1$	$-1$

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} $

203.1

−0.791071	+	0.611724i
−0.887586	−	0.460642i
−0.575617	−	0.817719i
−0.887586	+	0.460642i
0.925304	−	0.379225i
−0.193256	−	0.981148i
−0.251587	+	0.967835i
0.599822	+	0.800134i
0.842721	−	0.538351i
0.447313	+	0.894377i
0.999552	+	0.0299155i
0.599822	−	0.800134i
0.646600	+	0.762830i
−0.420357	+	0.907359i
−0.420357	−	0.907359i
0.772417	−	0.635116i
0.337330	+	0.941386i
−0.0149594	−	0.999888i
0.925304	+	0.379225i
−0.998210	−	0.0598042i

−0.691063

−

0.722795i

1.10130

0.0329607i

−0.0448648

0.998993i

−0.737244

−

0.818792i

0.753071

−

0.657939i

0.213567

0.0127951i

267.1

−0.963963

−

0.266037i

−0.246733

−

1.48580i

0.858449

0.512899i

−0.157435

1.49789i

−0.691063

−

0.722795i

−1.20038

0.409981i

339.1

0.858449

0.512899i

−0.254057

0.0867710i

0.473869

0.880596i

−0.262600

−

0.0558173i

−0.0448648

0.998993i

−0.734055

0.567635i

411.1

−0.963963

0.266037i

−0.246733

1.48580i

0.858449

−

0.512899i

−0.157435

−

1.49789i

−0.691063

0.722795i

−1.20038

−

0.409981i

427.1

−0.691063

0.722795i

−0.579195

−

0.937274i

−0.0448648

−

0.998993i

1.07772

0.229076i

0.753071

0.657939i

−0.0957026

0.191352i

443.1

−0.393025

−

0.919528i

−0.827762

−

0.461537i

−0.691063

0.722795i

−0.0990655

0.942546i

0.936235

0.351375i

−0.0535114

−

0.0865941i

531.1

−0.0448648

0.998993i

−0.784643

−

0.0470091i

−0.995974

−

0.0896393i

0.0821646

−

0.781744i

0.134233

−

0.990950i

−0.379392

−

0.0456236i

619.1

0.753071

0.657939i

1.13273

1.60916i

0.134233

0.990950i

−0.205697

1.95708i

−0.550897

0.834573i

−0.968970

2.70410i

883.1

−0.963963

−

0.266037i

−1.16337

0.956575i

0.858449

0.512899i

1.37593

−

0.612604i

−0.691063

−

0.722795i

0.245139

−

1.24455i

955.1

0.936235

−

0.351375i

−1.76803

0.917576i

0.753071

−

0.657939i

−1.33287

1.48031i

0.473869

−

0.880596i

1.70835

−

2.42688i

971.1

0.983930

−

0.178557i

0.0251627

−

0.0861293i

0.936235

−

0.351375i

0.00937930

−

0.0892381i

0.858449

−

0.512899i

0.835936

0.534016i

1027.1

0.753071

−

0.657939i

1.13273

−

1.60916i

0.134233

−

0.990950i

−0.205697

−

1.95708i

−0.550897

−

0.834573i

−0.968970

−

2.70410i

1115.1

0.473869

0.880596i

1.78392

0.731119i

−0.550897

0.834573i

0.201523

1.91736i

−0.995974

−

0.0896393i

1.93545

1.90671i

1131.1

0.858449

0.512899i

0.0518827

−

0.263406i

0.473869

0.880596i

0.179639

−

0.199510i

−0.0448648

0.998993i

0.858614

0.351893i

1171.1

0.858449

−

0.512899i

0.0518827

0.263406i

0.473869

−

0.880596i

0.179639

0.199510i

−0.0448648

−

0.998993i

0.858614

−

0.351893i

1219.1

−0.550897

−

0.834573i

0.431949

1.66167i

−0.393025

0.919528i

1.14883

−

1.27590i

0.983930

−

0.178557i

−1.70117

0.948526i

1235.1

0.473869

−

0.880596i

−1.52513

−

1.17936i

−0.550897

−

0.834573i

−1.76125

0.784158i

−0.995974

0.0896393i

0.683535

2.62951i

1307.1

−0.995974

−

0.0896393i

0.829029

−

1.10588i

0.983930

0.178557i

−0.924822

1.02712i

−0.963963

−

0.266037i

−0.255265

−

0.873746i

1347.1

−0.691063

−

0.722795i

−0.579195

0.937274i

−0.0448648

0.998993i

1.07772

−

0.229076i

0.753071

−

0.657939i

−0.0957026

−

0.191352i

1571.1

0.936235

−

0.351375i

1.67866

1.07237i

0.753071

−

0.657939i

1.94842

0.414149i

0.473869

−

0.880596i

1.24756

2.69291i

See all 48 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
473.bc	even	105	1	inner
3784.em	odd	210	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3784.1.em.b	yes	48
8.d	odd	2	1	CM	3784.1.em.b	yes	48
11.c	even	5	1		3784.1.em.a	✓	48
43.g	even	21	1		3784.1.em.a	✓	48
88.l	odd	10	1		3784.1.em.a	✓	48
344.be	odd	42	1		3784.1.em.a	✓	48
473.bc	even	105	1	inner	3784.1.em.b	yes	48
3784.em	odd	210	1	inner	3784.1.em.b	yes	48

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3784.1.em.a	✓	48	11.c	even	5	1
3784.1.em.a	✓	48	43.g	even	21	1
3784.1.em.a	✓	48	88.l	odd	10	1
3784.1.em.a	✓	48	344.be	odd	42	1
3784.1.em.b	yes	48	1.a	even	1	1	trivial
3784.1.em.b	yes	48	8.d	odd	2	1	CM
3784.1.em.b	yes	48	473.bc	even	105	1	inner
3784.1.em.b	yes	48	3784.em	odd	210	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{48} + 2 T_{3}^{47} + 4 T_{3}^{46} + 10 T_{3}^{45} + 43 T_{3}^{44} + 70 T_{3}^{43} + 149 T_{3}^{42} + \cdots + 1 $ T3^48 + 2*T3^47 + 4*T3^46 + 10*T3^45 + 43*T3^44 + 70*T3^43 + 149*T3^42 + 317*T3^41 + 942*T3^40 + 1660*T3^39 + 3327*T3^38 + 6282*T3^37 + 14943*T3^36 + 25863*T3^35 + 44748*T3^34 + 83179*T3^33 + 168786*T3^32 + 287032*T3^31 + 477152*T3^30 + 874293*T3^29 + 1415924*T3^28 + 2143422*T3^27 + 3064144*T3^26 + 4250465*T3^25 + 5540078*T3^24 + 6205646*T3^23 + 3648287*T3^22 - 2182045*T3^21 - 6526522*T3^20 - 12960478*T3^19 - 24048665*T3^18 - 34748216*T3^17 - 34308092*T3^16 - 20303748*T3^15 - 144829*T3^14 + 19650774*T3^13 + 31975795*T3^12 + 31615899*T3^11 + 25555209*T3^10 + 19120654*T3^9 + 10496598*T3^8 + 3633019*T3^7 + 977945*T3^6 + 233693*T3^5 + 39224*T3^4 + 3638*T3^3 + 284*T3^2 + 24*T3 + 1 acting on $S_{1}^{\mathrm{new}}(3784, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (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
$3$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 + 4T^46 + 10T^45 + 43T^44 + 70T^43 + 149T^42 + 317T^41 + 942T^40 + 1660T^39 + 3327T^38 + 6282T^37 + 14943T^36 + 25863T^35 + 44748T^34 + 83179T^33 + 168786T^32 + 287032T^31 + 477152T^30 + 874293T^29 + 1415924T^28 + 2143422T^27 + 3064144T^26 + 4250465T^25 + 5540078T^24 + 6205646T^23 + 3648287T^22 - 2182045T^21 - 6526522T^20 - 12960478T^19 - 24048665T^18 - 34748216T^17 - 34308092T^16 - 20303748T^15 - 144829T^14 + 19650774T^13 + 31975795T^12 + 31615899T^11 + 25555209T^10 + 19120654T^9 + 10496598T^8 + 3633019T^7 + 977945T^6 + 233693T^5 + 39224T^4 + 3638T^3 + 284T^2 + 24*T + 1
$5$	$ T^{48} $ T^48
$7$	$ T^{48} $ T^48
$11$	$ 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
$13$	$ T^{48} $ T^48
$17$	$ T^{48} - T^{47} + \cdots + 1 $ T^48 - T^47 + T^46 - 12T^45 + 7T^44 - 13T^43 + 43T^42 + 63T^41 - 56T^40 + 419T^39 - 1276T^38 + 1978T^37 - 5779T^36 + 7862T^35 - 11868T^34 + 22355T^33 + 4845T^32 - 128T^31 + 99415T^30 - 344244T^29 + 471231T^28 - 1515431T^27 + 2078724T^26 - 3509668T^25 + 6582615T^24 - 5531068T^23 + 13244586T^22 - 6942033T^21 + 17259106T^20 - 7280276T^19 + 15254526T^18 - 5641569T^17 + 9504628T^16 - 2902035T^15 + 4140785T^14 - 928274T^13 + 1049143T^12 - 66028T^11 + 309451T^10 + 187794T^9 - 50011T^8 - 93517T^7 + 66598T^6 + 22570T^5 - 10790T^4 - 2729T^3 + 985T^2 + 65*T + 1
$19$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 + 4T^46 + 14T^45 + 25T^44 + 50T^43 + 106T^42 + 153T^41 + 237T^40 + 286T^39 + 251T^38 - 400T^37 - 435T^36 - 281T^35 - 6855T^34 - 8892T^33 - 2809T^32 + 4640T^31 + 15850T^30 + 104316T^29 + 247873T^28 + 127356T^27 + 107602T^26 - 244413T^25 - 506636T^24 - 1456170T^23 - 1025877T^22 - 1391318T^21 + 1942148T^20 + 1633139T^19 + 2285107T^18 - 762328T^17 + 528973T^16 - 697436T^15 + 3110357T^14 - 146989T^13 + 1398825T^12 - 1011405T^11 - 128096T^10 - 176299T^9 + 249737T^8 + 115091T^7 - 36210T^6 - 25505T^5 - 419T^4 + 4583T^3 + 1001T^2 - 24*T + 1
$23$	$ T^{48} $ T^48
$29$	$ T^{48} $ T^48
$31$	$ T^{48} $ T^48
$37$	$ T^{48} $ T^48
$41$	$ 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
$43$	$ 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} $ T^48
$59$	$ T^{48} - 3 T^{47} + \cdots + 1 $ T^48 - 3T^47 + T^46 + 10T^45 - 27T^44 + 10T^43 + 69T^42 - 87T^41 - 146T^40 + 842T^39 - 991T^38 - 2074T^37 + 7205T^36 - 4493T^35 - 15501T^34 + 33032T^33 - 964T^32 - 89820T^31 + 190539T^30 - 168579T^29 - 187455T^28 + 761827T^27 - 902608T^26 + 293047T^25 + 537349T^24 - 544168T^23 + 328222T^22 - 1359034T^21 + 4020893T^20 - 8594416T^19 + 13735662T^18 - 16254232T^17 + 16818241T^16 - 15750374T^15 + 10853825T^14 - 4255557T^13 + 1517062T^12 - 2204357T^11 + 2031922T^10 - 633838T^9 - 122689T^8 + 156736T^7 - 43728T^6 - 4928T^5 + 3305T^4 - 1929T^3 + 1053T^2 + 35T + 1
$61$	$ T^{48} $ T^48
$67$	$ T^{48} - T^{47} + \cdots + 1 $ T^48 - T^47 - 4T^46 + 5T^45 - 4T^43 + 54T^42 - 83T^41 - 166T^40 + 326T^39 - 110T^38 + 40T^37 + 2426T^36 - 4176T^35 + 1276T^34 + 3579T^33 - 36239T^32 + 57054T^31 + 95795T^30 - 228790T^29 + 115384T^28 + 385320T^27 - 1928121T^26 + 1256250T^25 + 6202394T^24 - 8805690T^23 + 689904T^22 + 14637840T^21 - 19822361T^20 - 8874665T^19 + 42511835T^18 - 6959556T^17 - 53629094T^16 + 83319729T^15 - 16506914T^14 - 119500971T^13 + 119230196T^12 + 30147905T^11 + 1091980T^10 - 1088444T^9 - 276916T^8 - 9688T^7 + 12039T^6 + 436T^5 - 435T^4 - 110T^3 - 4T^2 + 4*T + 1
$71$	$ T^{48} $ T^48
$73$	$ T^{48} - T^{47} + \cdots + 1 $ T^48 - T^47 + T^46 - 5T^45 + T^43 - 6T^42 - 273T^41 + 259T^40 - 134T^39 + 1230T^38 + 200T^37 - 774T^36 + 1429T^35 + 26436T^34 - 24776T^33 - 839T^32 - 107466T^31 - 21020T^30 + 105870T^29 - 84436T^28 - 1078715T^27 + 1041639T^26 + 593900T^25 + 3511449T^24 + 169060T^23 - 4941981T^22 + 2093035T^21 + 16468344T^20 - 13692290T^19 - 16803780T^18 - 20511046T^17 + 15542681T^16 + 22522084T^15 - 40062304T^14 - 68697941T^13 + 117146306T^12 + 50912620T^11 + 15702290T^10 + 4196666T^9 + 1030019T^8 + 238017T^7 + 52514T^6 + 10551T^5 + 2020T^4 + 365T^3 + 61T^2 + 9*T + 1
$79$	$ T^{48} $ T^48
$83$	$ 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
$89$	$ T^{48} - T^{47} + \cdots + 1 $ T^48 - T^47 - 4T^46 - 2T^45 + 7T^44 + 17T^43 + 33T^42 - 27T^41 - 236T^40 + 39T^39 + 219T^38 + 1153T^37 + 501T^36 - 6598T^35 + 387T^34 + 12700T^33 - 6720T^32 - 16173T^31 + 85715T^30 - 162899T^29 + 65971T^28 + 333429T^27 - 753591T^26 + 22787T^25 + 2008260T^24 - 1242078T^23 - 3339184T^22 + 2272592T^21 + 3744826T^20 - 738691T^19 - 2490269T^18 - 2371259T^17 - 49967T^16 + 2831405T^15 + 2365800T^14 - 236469T^13 - 2395112T^12 - 2183723T^11 - 111719T^10 + 1410759T^9 + 1673634T^8 + 1121743T^7 + 532468T^6 + 186265T^5 + 49125T^4 + 8696T^3 + 605T^2 - 45T + 1
$97$	$ T^{48} - 3 T^{47} + \cdots + 1 $ T^48 - 3T^47 + T^46 + 3T^45 + T^44 + 10T^43 - 64T^42 + 109T^41 - 160T^40 + 303T^39 + 157T^38 - 625T^37 - 1636T^36 + 932T^35 + 12793T^34 - 17270T^33 - 7425T^32 - 18658T^31 + 114344T^30 - 47759T^29 - 23914T^28 - 169502T^27 + 294098T^26 + 18276T^25 + 287043T^24 - 363792T^23 - 722569T^22 - 78496T^21 - 755599T^20 + 3119853T^19 + 244604T^18 - 901461T^17 + 1978647T^16 - 5456216T^15 + 2044059T^14 - 294313T^13 + 305649T^12 + 993992T^11 + 15425T^10 - 354132T^9 - 112833T^8 - 30703T^7 + 45543T^6 + 7049T^5 + 2836T^4 + 3244T^3 + 1249T^2 + 49*T + 1
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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:	\( N \)	\(=\)	\( 3784 = 2^{3} \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3784.em (of order \(210\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{210}^{99} q^{2} + ( - \zeta_{210}^{11} + \zeta_{210}^{8}) q^{3} - \zeta_{210}^{93} q^{4} + ( - \zeta_{210}^{5} + \zeta_{210}^{2}) q^{6} - \zeta_{210}^{87} q^{8} + (\zeta_{210}^{22} + \cdots + \zeta_{210}^{16}) q^{9} +O(q^{10}) \) q - z^99 * q^2 + (-z^11 + z^8) * q^3 - z^93 * q^4 + (-z^5 + z^2) * q^6 - z^87 * q^8 + (z^22 - z^19 + z^16) * q^9	\( q - \zeta_{210}^{99} q^{2} + ( - \zeta_{210}^{11} + \zeta_{210}^{8}) q^{3} - \zeta_{210}^{93} q^{4} + ( - \zeta_{210}^{5} + \zeta_{210}^{2}) q^{6} - \zeta_{210}^{87} q^{8} + (\zeta_{210}^{22} + \cdots + \zeta_{210}^{16}) q^{9} + \cdots + (\zeta_{210}^{84} + \cdots + \zeta_{210}^{78}) q^{99} +O(q^{100}) \) q - z^99 * q^2 + (-z^11 + z^8) * q^3 - z^93 * q^4 + (-z^5 + z^2) * q^6 - z^87 * q^8 + (z^22 - z^19 + z^16) * q^9 + z^62 * q^11 + (z^104 - z^101) * q^12 - z^81 * q^16 + (-z^69 - z^23) * q^17 + (z^16 - z^13 + z^10) * q^18 + (z^60 - z^49) * q^19 + z^56 * q^22 + (z^98 - z^95) * q^24 - z^97 * q^25 + (-z^33 + z^30 - z^27 + z^24) * q^27 - z^75 * q^32 + (-z^73 + z^70) * q^33 + (-z^63 - z^17) * q^34 + (z^10 - z^7 + z^4) * q^36 + (z^54 - z^43) * q^38 + (z^88 + z^26) * q^41 - z * q^43 + z^50 * q^44 + (z^92 - z^89) * q^48 + z^56 * q^49 - z^91 * q^50 + (z^80 - z^77 + z^34 - z^31) * q^51 + (-z^27 + z^24 - z^21 + z^18) * q^54 + (-z^71 + z^68 + z^60 - z^57) * q^57 + (-z^101 + z^40) * q^59 - z^69 * q^64 + (-z^67 + z^64) * q^66 + (z^72 + z^58) * q^67 + (-z^57 - z^11) * q^68 + (-z^103 + z^4 - z) * q^72 + (-z^87 - z^59) * q^73 + (-z^3 + 1) * q^75 + (z^48 - z^37) * q^76 + (z^44 - z^41 + z^38 - z^35 + z^32) * q^81 + (z^82 + z^20) * q^82 + (-z^47 - z^35) * q^83 + z^100 * q^86 + z^44 * q^88 + (-z^31 - z^9) * q^89 + (z^86 - z^83) * q^96 + (-z^55 - z^53) * q^97 + z^50 * q^98 + (z^84 - z^81 + z^78) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{6} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) 48 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 3 * q^6 + 2 * q^8 - 3 * q^9	\( 48 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{6} + 2 q^{8} - 3 q^{9} - q^{11} - 2 q^{12} + 2 q^{16} + q^{17} + 2 q^{18} - 2 q^{19} + 6 q^{22} + 10 q^{24} - q^{25} - 2 q^{27} - 8 q^{32} - 25 q^{33} - 13 q^{34} + 9 q^{36} + q^{38} - 2 q^{41} - q^{43} + 4 q^{44} - 2 q^{48} + 6 q^{49} + 6 q^{50} + 8 q^{51} - 6 q^{54} - 8 q^{57} + 3 q^{59} + 2 q^{64} - 2 q^{66} + q^{67} + q^{68} - 3 q^{72} + q^{73} + 50 q^{75} + q^{76} - 28 q^{81} + 3 q^{82} - 25 q^{83} + 4 q^{86} - q^{88} + q^{89} - 2 q^{96} + 3 q^{97} + 4 q^{98} - 8 q^{99}+O(q^{100}) \) 48 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 3 * q^6 + 2 * q^8 - 3 * q^9 - q^11 - 2 * q^12 + 2 * q^16 + q^17 + 2 * q^18 - 2 * q^19 + 6 * q^22 + 10 * q^24 - q^25 - 2 * q^27 - 8 * q^32 - 25 * q^33 - 13 * q^34 + 9 * q^36 + q^38 - 2 * q^41 - q^43 + 4 * q^44 - 2 * q^48 + 6 * q^49 + 6 * q^50 + 8 * q^51 - 6 * q^54 - 8 * q^57 + 3 * q^59 + 2 * q^64 - 2 * q^66 + q^67 + q^68 - 3 * q^72 + q^73 + 50 * q^75 + q^76 - 28 * q^81 + 3 * q^82 - 25 * q^83 + 4 * q^86 - q^88 + q^89 - 2 * q^96 + 3 * q^97 + 4 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	3784.1.em.b

Level	$3784$
Weight	$1$
Character orbit	3784.em
Analytic conductor	$1.888$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{105}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.88846200780\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{105}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{105} - \cdots)\)

\(n\)	\(89\)	\(1377\)	\(1893\)	\(2839\)
\(\chi(n)\)	\(\zeta_{210}^{40}\)	\(-\zeta_{210}^{63}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 203.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
473.bc	even	105	1	inner
3784.em	odd	210	1	inner

$p$	$F_p(T)$
$2$	\( (T^{24} - T^{23} + T^{19} + \cdots + 1)^{2} \) (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
$3$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 + 4T^46 + 10T^45 + 43T^44 + 70T^43 + 149T^42 + 317T^41 + 942T^40 + 1660T^39 + 3327T^38 + 6282T^37 + 14943T^36 + 25863T^35 + 44748T^34 + 83179T^33 + 168786T^32 + 287032T^31 + 477152T^30 + 874293T^29 + 1415924T^28 + 2143422T^27 + 3064144T^26 + 4250465T^25 + 5540078T^24 + 6205646T^23 + 3648287T^22 - 2182045T^21 - 6526522T^20 - 12960478T^19 - 24048665T^18 - 34748216T^17 - 34308092T^16 - 20303748T^15 - 144829T^14 + 19650774T^13 + 31975795T^12 + 31615899T^11 + 25555209T^10 + 19120654T^9 + 10496598T^8 + 3633019T^7 + 977945T^6 + 233693T^5 + 39224T^4 + 3638T^3 + 284T^2 + 24*T + 1
$5$	\( T^{48} \) T^48
$7$	\( T^{48} \) T^48
$11$	\( T^{48} + T^{47} + \cdots + 1 \) 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
$13$	\( T^{48} \) T^48
$17$	\( T^{48} - T^{47} + \cdots + 1 \) T^48 - T^47 + T^46 - 12T^45 + 7T^44 - 13T^43 + 43T^42 + 63T^41 - 56T^40 + 419T^39 - 1276T^38 + 1978T^37 - 5779T^36 + 7862T^35 - 11868T^34 + 22355T^33 + 4845T^32 - 128T^31 + 99415T^30 - 344244T^29 + 471231T^28 - 1515431T^27 + 2078724T^26 - 3509668T^25 + 6582615T^24 - 5531068T^23 + 13244586T^22 - 6942033T^21 + 17259106T^20 - 7280276T^19 + 15254526T^18 - 5641569T^17 + 9504628T^16 - 2902035T^15 + 4140785T^14 - 928274T^13 + 1049143T^12 - 66028T^11 + 309451T^10 + 187794T^9 - 50011T^8 - 93517T^7 + 66598T^6 + 22570T^5 - 10790T^4 - 2729T^3 + 985T^2 + 65*T + 1
$19$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 + 4T^46 + 14T^45 + 25T^44 + 50T^43 + 106T^42 + 153T^41 + 237T^40 + 286T^39 + 251T^38 - 400T^37 - 435T^36 - 281T^35 - 6855T^34 - 8892T^33 - 2809T^32 + 4640T^31 + 15850T^30 + 104316T^29 + 247873T^28 + 127356T^27 + 107602T^26 - 244413T^25 - 506636T^24 - 1456170T^23 - 1025877T^22 - 1391318T^21 + 1942148T^20 + 1633139T^19 + 2285107T^18 - 762328T^17 + 528973T^16 - 697436T^15 + 3110357T^14 - 146989T^13 + 1398825T^12 - 1011405T^11 - 128096T^10 - 176299T^9 + 249737T^8 + 115091T^7 - 36210T^6 - 25505T^5 - 419T^4 + 4583T^3 + 1001T^2 - 24*T + 1
$23$	\( T^{48} \) T^48
$29$	\( T^{48} \) T^48
$31$	\( T^{48} \) T^48
$37$	\( T^{48} \) T^48
$41$	\( T^{48} + 2 T^{47} + \cdots + 1 \) 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
$43$	\( T^{48} + T^{47} + \cdots + 1 \) 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
$53$	\( T^{48} \) T^48
$59$	\( T^{48} - 3 T^{47} + \cdots + 1 \) T^48 - 3T^47 + T^46 + 10T^45 - 27T^44 + 10T^43 + 69T^42 - 87T^41 - 146T^40 + 842T^39 - 991T^38 - 2074T^37 + 7205T^36 - 4493T^35 - 15501T^34 + 33032T^33 - 964T^32 - 89820T^31 + 190539T^30 - 168579T^29 - 187455T^28 + 761827T^27 - 902608T^26 + 293047T^25 + 537349T^24 - 544168T^23 + 328222T^22 - 1359034T^21 + 4020893T^20 - 8594416T^19 + 13735662T^18 - 16254232T^17 + 16818241T^16 - 15750374T^15 + 10853825T^14 - 4255557T^13 + 1517062T^12 - 2204357T^11 + 2031922T^10 - 633838T^9 - 122689T^8 + 156736T^7 - 43728T^6 - 4928T^5 + 3305T^4 - 1929T^3 + 1053T^2 + 35T + 1
$61$	\( T^{48} \) T^48
$67$	\( T^{48} - T^{47} + \cdots + 1 \) T^48 - T^47 - 4T^46 + 5T^45 - 4T^43 + 54T^42 - 83T^41 - 166T^40 + 326T^39 - 110T^38 + 40T^37 + 2426T^36 - 4176T^35 + 1276T^34 + 3579T^33 - 36239T^32 + 57054T^31 + 95795T^30 - 228790T^29 + 115384T^28 + 385320T^27 - 1928121T^26 + 1256250T^25 + 6202394T^24 - 8805690T^23 + 689904T^22 + 14637840T^21 - 19822361T^20 - 8874665T^19 + 42511835T^18 - 6959556T^17 - 53629094T^16 + 83319729T^15 - 16506914T^14 - 119500971T^13 + 119230196T^12 + 30147905T^11 + 1091980T^10 - 1088444T^9 - 276916T^8 - 9688T^7 + 12039T^6 + 436T^5 - 435T^4 - 110T^3 - 4T^2 + 4*T + 1
$71$	\( T^{48} \) T^48
$73$	\( T^{48} - T^{47} + \cdots + 1 \) T^48 - T^47 + T^46 - 5T^45 + T^43 - 6T^42 - 273T^41 + 259T^40 - 134T^39 + 1230T^38 + 200T^37 - 774T^36 + 1429T^35 + 26436T^34 - 24776T^33 - 839T^32 - 107466T^31 - 21020T^30 + 105870T^29 - 84436T^28 - 1078715T^27 + 1041639T^26 + 593900T^25 + 3511449T^24 + 169060T^23 - 4941981T^22 + 2093035T^21 + 16468344T^20 - 13692290T^19 - 16803780T^18 - 20511046T^17 + 15542681T^16 + 22522084T^15 - 40062304T^14 - 68697941T^13 + 117146306T^12 + 50912620T^11 + 15702290T^10 + 4196666T^9 + 1030019T^8 + 238017T^7 + 52514T^6 + 10551T^5 + 2020T^4 + 365T^3 + 61T^2 + 9*T + 1
$79$	\( T^{48} \) T^48
$83$	\( T^{48} + 25 T^{47} + \cdots + 1 \) 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
$89$	\( T^{48} - T^{47} + \cdots + 1 \) T^48 - T^47 - 4T^46 - 2T^45 + 7T^44 + 17T^43 + 33T^42 - 27T^41 - 236T^40 + 39T^39 + 219T^38 + 1153T^37 + 501T^36 - 6598T^35 + 387T^34 + 12700T^33 - 6720T^32 - 16173T^31 + 85715T^30 - 162899T^29 + 65971T^28 + 333429T^27 - 753591T^26 + 22787T^25 + 2008260T^24 - 1242078T^23 - 3339184T^22 + 2272592T^21 + 3744826T^20 - 738691T^19 - 2490269T^18 - 2371259T^17 - 49967T^16 + 2831405T^15 + 2365800T^14 - 236469T^13 - 2395112T^12 - 2183723T^11 - 111719T^10 + 1410759T^9 + 1673634T^8 + 1121743T^7 + 532468T^6 + 186265T^5 + 49125T^4 + 8696T^3 + 605T^2 - 45T + 1
$97$	\( T^{48} - 3 T^{47} + \cdots + 1 \) T^48 - 3T^47 + T^46 + 3T^45 + T^44 + 10T^43 - 64T^42 + 109T^41 - 160T^40 + 303T^39 + 157T^38 - 625T^37 - 1636T^36 + 932T^35 + 12793T^34 - 17270T^33 - 7425T^32 - 18658T^31 + 114344T^30 - 47759T^29 - 23914T^28 - 169502T^27 + 294098T^26 + 18276T^25 + 287043T^24 - 363792T^23 - 722569T^22 - 78496T^21 - 755599T^20 + 3119853T^19 + 244604T^18 - 901461T^17 + 1978647T^16 - 5456216T^15 + 2044059T^14 - 294313T^13 + 305649T^12 + 993992T^11 + 15425T^10 - 354132T^9 - 112833T^8 - 30703T^7 + 45543T^6 + 7049T^5 + 2836T^4 + 3244T^3 + 1249T^2 + 49*T + 1
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