LMFDB - Newform orbit 3870.2.c.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3870,2,Mod(2321,3870)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3870.2321"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3870, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3870 = 2 \cdot 3^{2} \cdot 5 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3870.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,12,0,12,-12,0,0,12,0,-12,0,0,8,0,0,12,0,0,0,-12,0,0,0,0,12, 8,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.9021055822$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 6 x^{10} + 48 x^{9} + 53 x^{8} - 516 x^{7} + 540 x^{6} + 1520 x^{5} - 2672 x^{4} + \cdots + 31812 $ x^12 - 4x^11 - 6x^10 + 48x^9 + 53x^8 - 516x^7 + 540x^6 + 1520x^5 - 2672x^4 - 4152x^3 + 24376x^2 - 35424*x + 31812
Coefficient ring:	$\Z[a_1, \ldots, a_{43}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + q^{2} + q^{4} - q^{5} - \beta_{9} q^{7} + q^{8} - q^{10} + \beta_{3} q^{11} + ( - \beta_{4} + 1) q^{13} - \beta_{9} q^{14} + q^{16} + ( - \beta_{6} - \beta_{5}) q^{17} + (\beta_{9} - \beta_{5}) q^{19}+ \cdots + (\beta_{10} + \beta_{8} - \beta_{7} + \cdots - 5) q^{98}+O(q^{100}) $ q + q^2 + q^4 - q^5 - b9 * q^7 + q^8 - q^10 + b3 * q^11 + (-b4 + 1) * q^13 - b9 * q^14 + q^16 + (-b6 - b5) * q^17 + (b9 - b5) * q^19 - q^20 + b3 * q^22 + (b10 - b8 + b6 - b2) * q^23 + q^25 + (-b4 + 1) * q^26 - b9 * q^28 + (b11 - 1) * q^29 + (-b7 - b4 - 1) * q^31 + q^32 + (-b6 - b5) * q^34 + b9 * q^35 + (-b10 + b8 - b6 + b5) * q^37 + (b9 - b5) * q^38 - q^40 + (-b9 - b2) * q^41 + (b10 + b9 - b7 + 1) * q^43 + b3 * q^44 + (b10 - b8 + b6 - b2) * q^46 + (-b6 - b2) * q^47 + (b10 + b8 - b7 + 3b4 + b1 - 5) q^49 + q^50 + (-b4 + 1) * q^52 + (b10 - b8 - b6 + b5 + b3 - b2) * q^53 - b3 * q^55 - b9 * q^56 + (b11 - 1) * q^58 + (-2b9 + 2b5 - b3 - 2b2) q^59 + (-b9 + b6 - 3b5 - b3) q^61 + (-b7 - b4 - 1) * q^62 + q^64 + (b4 - 1) * q^65 + (-b7 + b4 - 1) * q^67 + (-b6 - b5) * q^68 + b9 * q^70 + (b11 + b10 + b8 + b7 - b4 + 2b1) q^71 + (-b10 - b9 + b8 + b3 + b2) * q^73 + (-b10 + b8 - b6 + b5) * q^74 + (b9 - b5) * q^76 + (b11 + b10 + b8 - b7 + b4 + 2b1 - 4) q^77 + (-b11 - 2b7 + 2b4 + b1 - 2) * q^79 - q^80 + (-b9 - b2) * q^82 + (b10 + 2b9 - b8 - b6 - 2b5 + b3) * q^83 + (b6 + b5) * q^85 + (b10 + b9 - b7 + 1) * q^86 + b3 * q^88 + (b11 - b4 + b1 - 1) * q^89 + (-3b9 - b6 + 3b5 - b2) * q^91 + (b10 - b8 + b6 - b2) * q^92 + (-b6 - b2) * q^94 + (-b9 + b5) * q^95 + (-2b1 + 2) q^97 + (b10 + b8 - b7 + 3b4 + b1 - 5) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} - 12 q^{5} + 12 q^{8} - 12 q^{10} + 8 q^{13} + 12 q^{16} - 12 q^{20} + 12 q^{25} + 8 q^{26} - 12 q^{29} - 12 q^{31} + 12 q^{32} - 12 q^{40} + 20 q^{43} - 36 q^{49} + 12 q^{50}+ \cdots - 36 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 - 12 * q^5 + 12 * q^8 - 12 * q^10 + 8 * q^13 + 12 * q^16 - 12 * q^20 + 12 * q^25 + 8 * q^26 - 12 * q^29 - 12 * q^31 + 12 * q^32 - 12 * q^40 + 20 * q^43 - 36 * q^49 + 12 * q^50 + 8 * q^52 - 12 * q^58 - 12 * q^62 + 12 * q^64 - 8 * q^65 - 4 * q^67 - 32 * q^77 - 8 * q^79 - 12 * q^80 + 20 * q^86 - 16 * q^89 + 24 * q^97 - 36 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4 x^{11} - 6 x^{10} + 48 x^{9} + 53 x^{8} - 516 x^{7} + 540 x^{6} + 1520 x^{5} - 2672 x^{4} + \cdots + 31812 $ x^12 - 4*x^11 - 6*x^10 + 48*x^9 + 53*x^8 - 516*x^7 + 540*x^6 + 1520*x^5 - 2672*x^4 - 4152*x^3 + 24376*x^2 - 35424*x + 31812 :

$\beta_{1}$	$=$	$ ( 34762162599 \nu^{11} - 816416881532 \nu^{10} + 212738608883 \nu^{9} + \cdots - 57\!\cdots\!66 ) / 696867707004854 $ (34762162599v^11 - 816416881532v^10 + 212738608883v^9 + 11494946170192v^8 + 1400892023404v^7 - 142361881344394v^6 + 69304391744508v^5 + 591970987104376v^4 - 399786382721980v^3 - 2307797467024564v^2 + 5204266114238076*v - 5737049490849666) / 696867707004854
$\beta_{2}$	$=$	$ ( - 1316518685 \nu^{11} + 5479243812 \nu^{10} + 5967302510 \nu^{9} - 57402754378 \nu^{8} + \cdots + 47849500483224 ) / 16791992939876 $ (-1316518685v^11 + 5479243812v^10 + 5967302510v^9 - 57402754378v^8 - 65179025735v^7 + 652073808578v^6 - 829125880808v^5 - 1136446457212v^4 + 2028044900774v^3 + 10795857744624v^2 - 42681951556168*v + 47849500483224) / 16791992939876
$\beta_{3}$	$=$	$ ( 177547854653 \nu^{11} - 446673727185 \nu^{10} - 1664197542692 \nu^{9} + \cdots - 23\!\cdots\!20 ) / 13\!\cdots\!08 $ (177547854653v^11 - 446673727185v^10 - 1664197542692v^9 + 6169769873956v^8 + 17366653990457v^7 - 64552980285245v^6 + 11080926408740v^5 + 294984085133980v^4 - 102409274433770v^3 - 331306724968722v^2 + 3682338404367412*v - 2366689539034920) / 1393735414009708
$\beta_{4}$	$=$	$ ( 110060999429 \nu^{11} - 522854104934 \nu^{10} - 767075299141 \nu^{9} + \cdots - 38\!\cdots\!56 ) / 696867707004854 $ (110060999429v^11 - 522854104934v^10 - 767075299141v^9 + 6431560342937v^8 + 4437839517516v^7 - 68533019769730v^6 + 45536349284150v^5 + 208902522137881v^4 - 291168323636612v^3 - 551410367949004v^2 + 1478936521173696*v - 3843976047863556) / 696867707004854
$\beta_{5}$	$=$	$ ( 6794 \nu^{11} + 6581 \nu^{10} - 135170 \nu^{9} + 78182 \nu^{8} + 1454678 \nu^{7} + \cdots + 115086444 ) / 42644404 $ (6794v^11 + 6581v^10 - 135170v^9 + 78182v^8 + 1454678v^7 - 778367v^6 - 8052486v^5 + 11558450v^4 + 26719624v^3 - 22132394v^2 + 1251064*v + 115086444) / 42644404
$\beta_{6}$	$=$	$ ( - 196030836203 \nu^{11} + 675090792523 \nu^{10} + 1455404905837 \nu^{9} + \cdots + 50\!\cdots\!10 ) / 696867707004854 $ (-196030836203v^11 + 675090792523v^10 + 1455404905837v^9 - 8218408162829v^8 - 14132898242762v^7 + 88774954588365v^6 - 59566768936830v^5 - 285917030202947v^4 + 479754788657162v^3 + 865671743634398v^2 - 4506834609018652*v + 5034835740035310) / 696867707004854
$\beta_{7}$	$=$	$ ( 257792405929 \nu^{11} - 602128958753 \nu^{10} - 2828301166157 \nu^{9} + \cdots - 52\!\cdots\!10 ) / 696867707004854 $ (257792405929v^11 - 602128958753v^10 - 2828301166157v^9 + 7136154092493v^8 + 33113193197564v^7 - 106195656508159v^6 - 35999190888278v^5 + 463464578390605v^4 + 18243024427972v^3 - 2493526730958478v^2 + 4625034046090176*v - 5283682123247110) / 696867707004854
$\beta_{8}$	$=$	$ ( 75785676316 \nu^{11} - 279100909404 \nu^{10} - 807210833708 \nu^{9} + \cdots - 11\!\cdots\!48 ) / 199105059144244 $ (75785676316v^11 - 279100909404v^10 - 807210833708v^9 + 4145174236425v^8 + 6018135866577v^7 - 42739719756013v^6 + 18535741295791v^5 + 181505630997350v^4 - 215908485455434v^3 - 584464755676314v^2 + 1504602670254630*v - 1101444791806348) / 199105059144244
$\beta_{9}$	$=$	$ ( - 594215435388 \nu^{11} + 1525852485556 \nu^{10} + 7155369565217 \nu^{9} + \cdots + 634134427493820 ) / 13\!\cdots\!08 $ (-594215435388v^11 + 1525852485556v^10 + 7155369565217v^9 - 16697031139379v^8 - 85841444097075v^7 + 182122412093829v^6 + 295676101455792v^5 - 684732843990360v^4 - 1042446614628598v^3 + 3111775104699190v^2 - 5009440990700656*v + 634134427493820) / 1393735414009708
$\beta_{10}$	$=$	$ ( 756489289712 \nu^{11} - 1908197944226 \nu^{10} - 6412438400100 \nu^{9} + \cdots + 14443337292956 ) / 13\!\cdots\!08 $ (756489289712v^11 - 1908197944226v^10 - 6412438400100v^9 + 24199783733761v^8 + 69004612674757v^7 - 254027670274351v^6 - 18754129695625v^5 + 1146325070476762v^4 + 68885910567454v^3 - 3885057802305398v^2 + 7604883032773542*v + 14443337292956) / 1393735414009708
$\beta_{11}$	$=$	$ ( - 218065104578 \nu^{11} + 558439777238 \nu^{10} + 2382143949830 \nu^{9} + \cdots + 23\!\cdots\!89 ) / 348433853502427 $ (-218065104578v^11 + 558439777238v^10 + 2382143949830v^9 - 7486527848006v^8 - 24753913784766v^7 + 92579765225580v^6 + 24099681651226v^5 - 436513064856935v^4 + 103503423439084v^3 + 1936961522541060v^2 - 3818721203239524*v + 2367684171009289) / 348433853502427

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -2\beta_{11} - \beta_{10} - \beta_{8} - 2\beta_{7} + 2\beta_{3} - 2\beta_{2} + 2 ) / 6 $ (-2b11 - b10 - b8 - 2b7 + 2b3 - 2b2 + 2) / 6
$\nu^{2}$	$=$	$ ( \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + 3\beta_{4} + 8\beta_{3} + 4\beta_{2} + 14 ) / 6 $ (b11 - b10 - b8 + b7 + 3b4 + 8b3 + 4*b2 + 14) / 6
$\nu^{3}$	$=$	$ ( 9\beta_{6} + 3\beta_{4} + 10\beta_{3} - 10\beta_{2} - 3 ) / 3 $ (9b6 + 3b4 + 10b3 - 10b2 - 3) / 3
$\nu^{4}$	$=$	$ ( - 5 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} - 2 \beta_{7} + 12 \beta_{6} - 12 \beta_{5} - 6 \beta_{4} + \cdots - 67 ) / 3 $ (-5b11 + 2b10 + 2b8 - 2b7 + 12b6 - 12b5 - 6b4 + 28b3 + 20*b2 - 67) / 3
$\nu^{5}$	$=$	$ ( 59 \beta_{11} + 16 \beta_{10} + 46 \beta_{8} + 62 \beta_{7} + 45 \beta_{6} - 15 \beta_{5} - 30 \beta_{4} + \cdots - 14 ) / 3 $ (59b11 + 16b10 + 46b8 + 62b7 + 45b6 - 15b5 - 30b4 + 49b3 - 19b2 - 3b1 - 14) / 3
$\nu^{6}$	$=$	$ ( - 12 \beta_{11} + 39 \beta_{10} - 36 \beta_{9} + 75 \beta_{8} - 63 \beta_{7} + 42 \beta_{6} + \cdots - 1065 ) / 3 $ (-12b11 + 39b10 - 36b9 + 75b8 - 63b7 + 42b6 - 78b5 - 183b4 + 34b3 + 2b2 - 6*b1 - 1065) / 3
$\nu^{7}$	$=$	$ ( 649 \beta_{11} + 215 \beta_{10} - 84 \beta_{9} + 341 \beta_{8} + 568 \beta_{7} - 231 \beta_{6} + \cdots - 478 ) / 3 $ (649b11 + 215b10 - 84b9 + 341b8 + 568b7 - 231b6 - 21b5 - 588b4 - 239b3 + 365b2 + 81*b1 - 478) / 3
$\nu^{8}$	$=$	$ ( 635 \beta_{11} + 550 \beta_{10} - 48 \beta_{9} + 598 \beta_{8} - 334 \beta_{7} - 600 \beta_{6} + \cdots - 5489 ) / 3 $ (635b11 + 550b10 - 48b9 + 598b8 - 334b7 - 600b6 + 744b5 - 1212b4 - 1784b3 - 1192b2 + 246*b1 - 5489) / 3
$\nu^{9}$	$=$	$ ( 1965 \beta_{11} + 2355 \beta_{10} + 396 \beta_{9} - 975 \beta_{8} + 726 \beta_{7} - 5301 \beta_{6} + \cdots - 4500 ) / 3 $ (1965b11 + 2355b10 + 396b9 - 975b8 + 726b7 - 5301b6 + 1251b5 - 3036b4 - 6335b3 + 3113b2 + 1263*b1 - 4500) / 3
$\nu^{10}$	$=$	$ ( 4849 \beta_{11} + 3374 \beta_{10} + 6252 \beta_{9} - 2398 \beta_{8} + 2032 \beta_{7} - 10974 \beta_{6} + \cdots + 24131 ) / 3 $ (4849b11 + 3374b10 + 6252b9 - 2398b8 + 2032b7 - 10974b6 + 15186b5 + 1494b4 - 20462b3 - 10066b2 + 2106*b1 + 24131) / 3
$\nu^{11}$	$=$	$ ( - 32995 \beta_{11} + 9715 \beta_{10} + 16236 \beta_{9} - 38795 \beta_{8} - 33814 \beta_{7} + \cdots + 11584 ) / 3 $ (-32995b11 + 9715b10 + 16236b9 - 38795b8 - 33814b7 - 36135b6 + 18645b5 + 24768b4 - 46661b3 - 3037b2 - 1329*b1 + 11584) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3870\mathbb{Z}\right)^\times$.

$n$	$1721$	$1981$	$3097$
$\chi(n)$	$-1$	$-1$	$1$

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} $

2321.1

1.15705	−	1.41421i
−2.49192	+	1.41421i
2.33486	+	1.41421i
0.522916	+	1.41421i
2.74979	−	1.41421i
−2.27271	−	1.41421i
−2.27271	+	1.41421i
2.74979	+	1.41421i
0.522916	−	1.41421i
2.33486	−	1.41421i
−2.49192	−	1.41421i
1.15705	+	1.41421i

1.00000

−1.00000

−

5.14111i

1.00000

−1.00000

2321.2

1.00000

−1.00000

−

4.26870i

1.00000

−1.00000

2321.3

1.00000

−1.00000

−

2.80426i

1.00000

−1.00000

2321.4

1.00000

−1.00000

−

1.92901i

1.00000

−1.00000

2321.5

1.00000

−1.00000

−

1.84349i

1.00000

−1.00000

2321.6

1.00000

−1.00000

−

0.603153i

1.00000

−1.00000

2321.7

1.00000

−1.00000

0.603153i

1.00000

−1.00000

2321.8

1.00000

−1.00000

1.84349i

1.00000

−1.00000

2321.9

1.00000

−1.00000

1.92901i

1.00000

−1.00000

2321.10

1.00000

−1.00000

2.80426i

1.00000

−1.00000

2321.11

1.00000

−1.00000

4.26870i

1.00000

−1.00000

2321.12

1.00000

−1.00000

5.14111i

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
129.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3870.2.c.f	yes	12
3.b	odd	2	1		3870.2.c.e	✓	12
43.b	odd	2	1		3870.2.c.e	✓	12
129.d	even	2	1	inner	3870.2.c.f	yes	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3870.2.c.e	✓	12	3.b	odd	2	1
3870.2.c.e	✓	12	43.b	odd	2	1
3870.2.c.f	yes	12	1.a	even	1	1	trivial
3870.2.c.f	yes	12	129.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3870, [\chi])$:

$ T_{7}^{12} + 60T_{7}^{10} + 1241T_{7}^{8} + 10824T_{7}^{6} + 41272T_{7}^{4} + 61536T_{7}^{2} + 17424 $

T7^12 + 60*T7^10 + 1241*T7^8 + 10824*T7^6 + 41272*T7^4 + 61536*T7^2 + 17424

$ T_{29}^{6} + 6T_{29}^{5} - 55T_{29}^{4} - 192T_{29}^{3} + 1024T_{29}^{2} + 192T_{29} - 1152 $

T29^6 + 6*T29^5 - 55*T29^4 - 192*T29^3 + 1024*T29^2 + 192*T29 - 1152

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T - 1)^{12} $ (T - 1)^12
$3$	$ T^{12} $ T^12
$5$	$ (T + 1)^{12} $ (T + 1)^12
$7$	$ T^{12} + 60 T^{10} + \cdots + 17424 $ T^12 + 60T^10 + 1241T^8 + 10824T^6 + 41272T^4 + 61536*T^2 + 17424
$11$	$ T^{12} + 72 T^{10} + \cdots + 389376 $ T^12 + 72T^10 + 1880T^8 + 22368T^6 + 130576T^4 + 364800*T^2 + 389376
$13$	$ (T^{6} - 4 T^{5} - 23 T^{4} + \cdots + 88)^{2} $ (T^6 - 4T^5 - 23T^4 + 80T^3 + 56T^2 - 216*T + 88)^2
$17$	$ T^{12} + 120 T^{10} + \cdots + 278784 $ T^12 + 120T^10 + 4712T^8 + 72864T^6 + 403600T^4 + 664320*T^2 + 278784
$19$	$ T^{12} + 84 T^{10} + \cdots + 576 $ T^12 + 84T^10 + 1673T^8 + 9096T^6 + 16624T^4 + 6528*T^2 + 576
$23$	$ T^{12} + 168 T^{10} + \cdots + 2985984 $ T^12 + 168T^10 + 9968T^8 + 241152T^6 + 1992256T^4 + 5004288*T^2 + 2985984
$29$	$ (T^{6} + 6 T^{5} + \cdots - 1152)^{2} $ (T^6 + 6T^5 - 55T^4 - 192T^3 + 1024T^2 + 192*T - 1152)^2
$31$	$ (T^{6} + 6 T^{5} + \cdots - 3488)^{2} $ (T^6 + 6T^5 - 91T^4 - 388T^3 + 2320T^2 + 4448*T - 3488)^2
$37$	$ T^{12} + \cdots + 160579584 $ T^12 + 248T^10 + 19512T^8 + 681056T^6 + 11197840T^4 + 79027200*T^2 + 160579584
$41$	$ T^{12} + 100 T^{10} + \cdots + 1296 $ T^12 + 100T^10 + 3153T^8 + 32632T^6 + 102280T^4 + 53280*T^2 + 1296
$43$	$ T^{12} + \cdots + 6321363049 $ T^12 - 20T^11 + 102T^10 + 532T^9 - 6249T^8 - 12520T^7 + 350388T^6 - 538360T^5 - 11554401T^4 + 42297724T^3 + 348717702T^2 - 2940168860*T + 6321363049
$47$	$ T^{12} + 152 T^{10} + \cdots + 63489024 $ T^12 + 152T^10 + 8832T^8 + 249728T^6 + 3598144T^4 + 24794112*T^2 + 63489024
$53$	$ T^{12} + \cdots + 160579584 $ T^12 + 436T^10 + 71076T^8 + 5278336T^6 + 168262144T^4 + 1490706432*T^2 + 160579584
$59$	$ T^{12} + \cdots + 375366979584 $ T^12 + 600T^10 + 139976T^8 + 16317792T^6 + 1003956112T^4 + 30956196864*T^2 + 375366979584
$61$	$ T^{12} + \cdots + 22647842064 $ T^12 + 500T^10 + 95745T^8 + 8785112T^6 + 389888200T^4 + 7067750304*T^2 + 22647842064
$67$	$ (T^{6} + 2 T^{5} + \cdots - 2288)^{2} $ (T^6 + 2T^5 - 55T^4 - 28T^3 + 704T^2 + 80*T - 2288)^2
$71$	$ (T^{6} - 252 T^{4} + \cdots - 228096)^{2} $ (T^6 - 252T^4 + 16128T^2 - 6912*T - 228096)^2
$73$	$ T^{12} + 284 T^{10} + \cdots + 627264 $ T^12 + 284T^10 + 20193T^8 + 525848T^6 + 4795696T^4 + 8673408*T^2 + 627264
$79$	$ (T^{6} + 4 T^{5} + \cdots - 388224)^{2} $ (T^6 + 4T^5 - 359T^4 - 816T^3 + 31968T^2 + 55872*T - 388224)^2
$83$	$ T^{12} + \cdots + 125619264 $ T^12 + 516T^10 + 94796T^8 + 7447584T^6 + 238932016T^4 + 2375040576*T^2 + 125619264
$89$	$ (T^{6} + 8 T^{5} + \cdots + 3168)^{2} $ (T^6 + 8T^5 - 108T^4 - 1192T^3 - 3368T^2 - 1632*T + 3168)^2
$97$	$ (T^{6} - 12 T^{5} + \cdots - 32768)^{2} $ (T^6 - 12T^5 - 232T^4 + 2176T^3 + 10240T^2 - 8192*T - 32768)^2
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3870 = 2 \cdot 3^{2} \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3870.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + q^{2} + q^{4} - q^{5} - \beta_{9} q^{7} + q^{8} - q^{10} + \beta_{3} q^{11} + ( - \beta_{4} + 1) q^{13} - \beta_{9} q^{14} + q^{16} + ( - \beta_{6} - \beta_{5}) q^{17} + (\beta_{9} - \beta_{5}) q^{19}+ \cdots + (\beta_{10} + \beta_{8} - \beta_{7} + \cdots - 5) q^{98}+O(q^{100}) \) q + q^2 + q^4 - q^5 - b9 * q^7 + q^8 - q^10 + b3 * q^11 + (-b4 + 1) * q^13 - b9 * q^14 + q^16 + (-b6 - b5) * q^17 + (b9 - b5) * q^19 - q^20 + b3 * q^22 + (b10 - b8 + b6 - b2) * q^23 + q^25 + (-b4 + 1) * q^26 - b9 * q^28 + (b11 - 1) * q^29 + (-b7 - b4 - 1) * q^31 + q^32 + (-b6 - b5) * q^34 + b9 * q^35 + (-b10 + b8 - b6 + b5) * q^37 + (b9 - b5) * q^38 - q^40 + (-b9 - b2) * q^41 + (b10 + b9 - b7 + 1) * q^43 + b3 * q^44 + (b10 - b8 + b6 - b2) * q^46 + (-b6 - b2) * q^47 + (b10 + b8 - b7 + 3b4 + b1 - 5) q^49 + q^50 + (-b4 + 1) * q^52 + (b10 - b8 - b6 + b5 + b3 - b2) * q^53 - b3 * q^55 - b9 * q^56 + (b11 - 1) * q^58 + (-2b9 + 2b5 - b3 - 2b2) q^59 + (-b9 + b6 - 3b5 - b3) q^61 + (-b7 - b4 - 1) * q^62 + q^64 + (b4 - 1) * q^65 + (-b7 + b4 - 1) * q^67 + (-b6 - b5) * q^68 + b9 * q^70 + (b11 + b10 + b8 + b7 - b4 + 2b1) q^71 + (-b10 - b9 + b8 + b3 + b2) * q^73 + (-b10 + b8 - b6 + b5) * q^74 + (b9 - b5) * q^76 + (b11 + b10 + b8 - b7 + b4 + 2b1 - 4) q^77 + (-b11 - 2b7 + 2b4 + b1 - 2) * q^79 - q^80 + (-b9 - b2) * q^82 + (b10 + 2b9 - b8 - b6 - 2b5 + b3) * q^83 + (b6 + b5) * q^85 + (b10 + b9 - b7 + 1) * q^86 + b3 * q^88 + (b11 - b4 + b1 - 1) * q^89 + (-3b9 - b6 + 3b5 - b2) * q^91 + (b10 - b8 + b6 - b2) * q^92 + (-b6 - b2) * q^94 + (-b9 + b5) * q^95 + (-2b1 + 2) q^97 + (b10 + b8 - b7 + 3b4 + b1 - 5) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 12 q^{5} + 12 q^{8} - 12 q^{10} + 8 q^{13} + 12 q^{16} - 12 q^{20} + 12 q^{25} + 8 q^{26} - 12 q^{29} - 12 q^{31} + 12 q^{32} - 12 q^{40} + 20 q^{43} - 36 q^{49} + 12 q^{50}+ \cdots - 36 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 - 12 * q^5 + 12 * q^8 - 12 * q^10 + 8 * q^13 + 12 * q^16 - 12 * q^20 + 12 * q^25 + 8 * q^26 - 12 * q^29 - 12 * q^31 + 12 * q^32 - 12 * q^40 + 20 * q^43 - 36 * q^49 + 12 * q^50 + 8 * q^52 - 12 * q^58 - 12 * q^62 + 12 * q^64 - 8 * q^65 - 4 * q^67 - 32 * q^77 - 8 * q^79 - 12 * q^80 + 20 * q^86 - 16 * q^89 + 24 * q^97 - 36 * q^98

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	3870.2.c.f

Level	$3870$
Weight	$2$
Character orbit	3870.c
Analytic conductor	$30.902$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(30.9021055822\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4 x^{11} - 6 x^{10} + 48 x^{9} + 53 x^{8} - 516 x^{7} + 540 x^{6} + 1520 x^{5} - 2672 x^{4} + \cdots + 31812 \) x^12 - 4x^11 - 6x^10 + 48x^9 + 53x^8 - 516x^7 + 540x^6 + 1520x^5 - 2672x^4 - 4152x^3 + 24376x^2 - 35424*x + 31812
Coefficient ring:	\(\Z[a_1, \ldots, a_{43}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 34762162599 \nu^{11} - 816416881532 \nu^{10} + 212738608883 \nu^{9} + \cdots - 57\!\cdots\!66 ) / 696867707004854 \) (34762162599v^11 - 816416881532v^10 + 212738608883v^9 + 11494946170192v^8 + 1400892023404v^7 - 142361881344394v^6 + 69304391744508v^5 + 591970987104376v^4 - 399786382721980v^3 - 2307797467024564v^2 + 5204266114238076*v - 5737049490849666) / 696867707004854
\(\beta_{2}\)	\(=\)	\( ( - 1316518685 \nu^{11} + 5479243812 \nu^{10} + 5967302510 \nu^{9} - 57402754378 \nu^{8} + \cdots + 47849500483224 ) / 16791992939876 \) (-1316518685v^11 + 5479243812v^10 + 5967302510v^9 - 57402754378v^8 - 65179025735v^7 + 652073808578v^6 - 829125880808v^5 - 1136446457212v^4 + 2028044900774v^3 + 10795857744624v^2 - 42681951556168*v + 47849500483224) / 16791992939876
\(\beta_{3}\)	\(=\)	\( ( 177547854653 \nu^{11} - 446673727185 \nu^{10} - 1664197542692 \nu^{9} + \cdots - 23\!\cdots\!20 ) / 13\!\cdots\!08 \) (177547854653v^11 - 446673727185v^10 - 1664197542692v^9 + 6169769873956v^8 + 17366653990457v^7 - 64552980285245v^6 + 11080926408740v^5 + 294984085133980v^4 - 102409274433770v^3 - 331306724968722v^2 + 3682338404367412*v - 2366689539034920) / 1393735414009708
\(\beta_{4}\)	\(=\)	\( ( 110060999429 \nu^{11} - 522854104934 \nu^{10} - 767075299141 \nu^{9} + \cdots - 38\!\cdots\!56 ) / 696867707004854 \) (110060999429v^11 - 522854104934v^10 - 767075299141v^9 + 6431560342937v^8 + 4437839517516v^7 - 68533019769730v^6 + 45536349284150v^5 + 208902522137881v^4 - 291168323636612v^3 - 551410367949004v^2 + 1478936521173696*v - 3843976047863556) / 696867707004854
\(\beta_{5}\)	\(=\)	\( ( 6794 \nu^{11} + 6581 \nu^{10} - 135170 \nu^{9} + 78182 \nu^{8} + 1454678 \nu^{7} + \cdots + 115086444 ) / 42644404 \) (6794v^11 + 6581v^10 - 135170v^9 + 78182v^8 + 1454678v^7 - 778367v^6 - 8052486v^5 + 11558450v^4 + 26719624v^3 - 22132394v^2 + 1251064*v + 115086444) / 42644404
\(\beta_{6}\)	\(=\)	\( ( - 196030836203 \nu^{11} + 675090792523 \nu^{10} + 1455404905837 \nu^{9} + \cdots + 50\!\cdots\!10 ) / 696867707004854 \) (-196030836203v^11 + 675090792523v^10 + 1455404905837v^9 - 8218408162829v^8 - 14132898242762v^7 + 88774954588365v^6 - 59566768936830v^5 - 285917030202947v^4 + 479754788657162v^3 + 865671743634398v^2 - 4506834609018652*v + 5034835740035310) / 696867707004854
\(\beta_{7}\)	\(=\)	\( ( 257792405929 \nu^{11} - 602128958753 \nu^{10} - 2828301166157 \nu^{9} + \cdots - 52\!\cdots\!10 ) / 696867707004854 \) (257792405929v^11 - 602128958753v^10 - 2828301166157v^9 + 7136154092493v^8 + 33113193197564v^7 - 106195656508159v^6 - 35999190888278v^5 + 463464578390605v^4 + 18243024427972v^3 - 2493526730958478v^2 + 4625034046090176*v - 5283682123247110) / 696867707004854
\(\beta_{8}\)	\(=\)	\( ( 75785676316 \nu^{11} - 279100909404 \nu^{10} - 807210833708 \nu^{9} + \cdots - 11\!\cdots\!48 ) / 199105059144244 \) (75785676316v^11 - 279100909404v^10 - 807210833708v^9 + 4145174236425v^8 + 6018135866577v^7 - 42739719756013v^6 + 18535741295791v^5 + 181505630997350v^4 - 215908485455434v^3 - 584464755676314v^2 + 1504602670254630*v - 1101444791806348) / 199105059144244
\(\beta_{9}\)	\(=\)	\( ( - 594215435388 \nu^{11} + 1525852485556 \nu^{10} + 7155369565217 \nu^{9} + \cdots + 634134427493820 ) / 13\!\cdots\!08 \) (-594215435388v^11 + 1525852485556v^10 + 7155369565217v^9 - 16697031139379v^8 - 85841444097075v^7 + 182122412093829v^6 + 295676101455792v^5 - 684732843990360v^4 - 1042446614628598v^3 + 3111775104699190v^2 - 5009440990700656*v + 634134427493820) / 1393735414009708
\(\beta_{10}\)	\(=\)	\( ( 756489289712 \nu^{11} - 1908197944226 \nu^{10} - 6412438400100 \nu^{9} + \cdots + 14443337292956 ) / 13\!\cdots\!08 \) (756489289712v^11 - 1908197944226v^10 - 6412438400100v^9 + 24199783733761v^8 + 69004612674757v^7 - 254027670274351v^6 - 18754129695625v^5 + 1146325070476762v^4 + 68885910567454v^3 - 3885057802305398v^2 + 7604883032773542*v + 14443337292956) / 1393735414009708
\(\beta_{11}\)	\(=\)	\( ( - 218065104578 \nu^{11} + 558439777238 \nu^{10} + 2382143949830 \nu^{9} + \cdots + 23\!\cdots\!89 ) / 348433853502427 \) (-218065104578v^11 + 558439777238v^10 + 2382143949830v^9 - 7486527848006v^8 - 24753913784766v^7 + 92579765225580v^6 + 24099681651226v^5 - 436513064856935v^4 + 103503423439084v^3 + 1936961522541060v^2 - 3818721203239524*v + 2367684171009289) / 348433853502427

\(\nu\)	\(=\)	\( ( -2\beta_{11} - \beta_{10} - \beta_{8} - 2\beta_{7} + 2\beta_{3} - 2\beta_{2} + 2 ) / 6 \) (-2b11 - b10 - b8 - 2b7 + 2b3 - 2b2 + 2) / 6
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + 3\beta_{4} + 8\beta_{3} + 4\beta_{2} + 14 ) / 6 \) (b11 - b10 - b8 + b7 + 3b4 + 8b3 + 4*b2 + 14) / 6
\(\nu^{3}\)	\(=\)	\( ( 9\beta_{6} + 3\beta_{4} + 10\beta_{3} - 10\beta_{2} - 3 ) / 3 \) (9b6 + 3b4 + 10b3 - 10b2 - 3) / 3
\(\nu^{4}\)	\(=\)	\( ( - 5 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} - 2 \beta_{7} + 12 \beta_{6} - 12 \beta_{5} - 6 \beta_{4} + \cdots - 67 ) / 3 \) (-5b11 + 2b10 + 2b8 - 2b7 + 12b6 - 12b5 - 6b4 + 28b3 + 20*b2 - 67) / 3
\(\nu^{5}\)	\(=\)	\( ( 59 \beta_{11} + 16 \beta_{10} + 46 \beta_{8} + 62 \beta_{7} + 45 \beta_{6} - 15 \beta_{5} - 30 \beta_{4} + \cdots - 14 ) / 3 \) (59b11 + 16b10 + 46b8 + 62b7 + 45b6 - 15b5 - 30b4 + 49b3 - 19b2 - 3b1 - 14) / 3
\(\nu^{6}\)	\(=\)	\( ( - 12 \beta_{11} + 39 \beta_{10} - 36 \beta_{9} + 75 \beta_{8} - 63 \beta_{7} + 42 \beta_{6} + \cdots - 1065 ) / 3 \) (-12b11 + 39b10 - 36b9 + 75b8 - 63b7 + 42b6 - 78b5 - 183b4 + 34b3 + 2b2 - 6*b1 - 1065) / 3
\(\nu^{7}\)	\(=\)	\( ( 649 \beta_{11} + 215 \beta_{10} - 84 \beta_{9} + 341 \beta_{8} + 568 \beta_{7} - 231 \beta_{6} + \cdots - 478 ) / 3 \) (649b11 + 215b10 - 84b9 + 341b8 + 568b7 - 231b6 - 21b5 - 588b4 - 239b3 + 365b2 + 81*b1 - 478) / 3
\(\nu^{8}\)	\(=\)	\( ( 635 \beta_{11} + 550 \beta_{10} - 48 \beta_{9} + 598 \beta_{8} - 334 \beta_{7} - 600 \beta_{6} + \cdots - 5489 ) / 3 \) (635b11 + 550b10 - 48b9 + 598b8 - 334b7 - 600b6 + 744b5 - 1212b4 - 1784b3 - 1192b2 + 246*b1 - 5489) / 3
\(\nu^{9}\)	\(=\)	\( ( 1965 \beta_{11} + 2355 \beta_{10} + 396 \beta_{9} - 975 \beta_{8} + 726 \beta_{7} - 5301 \beta_{6} + \cdots - 4500 ) / 3 \) (1965b11 + 2355b10 + 396b9 - 975b8 + 726b7 - 5301b6 + 1251b5 - 3036b4 - 6335b3 + 3113b2 + 1263*b1 - 4500) / 3
\(\nu^{10}\)	\(=\)	\( ( 4849 \beta_{11} + 3374 \beta_{10} + 6252 \beta_{9} - 2398 \beta_{8} + 2032 \beta_{7} - 10974 \beta_{6} + \cdots + 24131 ) / 3 \) (4849b11 + 3374b10 + 6252b9 - 2398b8 + 2032b7 - 10974b6 + 15186b5 + 1494b4 - 20462b3 - 10066b2 + 2106*b1 + 24131) / 3
\(\nu^{11}\)	\(=\)	\( ( - 32995 \beta_{11} + 9715 \beta_{10} + 16236 \beta_{9} - 38795 \beta_{8} - 33814 \beta_{7} + \cdots + 11584 ) / 3 \) (-32995b11 + 9715b10 + 16236b9 - 38795b8 - 33814b7 - 36135b6 + 18645b5 + 24768b4 - 46661b3 - 3037b2 - 1329*b1 + 11584) / 3

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 2321.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T - 1)^{12} \) (T - 1)^12
$3$	\( T^{12} \) T^12
$5$	\( (T + 1)^{12} \) (T + 1)^12
$7$	\( T^{12} + 60 T^{10} + \cdots + 17424 \) T^12 + 60T^10 + 1241T^8 + 10824T^6 + 41272T^4 + 61536*T^2 + 17424
$11$	\( T^{12} + 72 T^{10} + \cdots + 389376 \) T^12 + 72T^10 + 1880T^8 + 22368T^6 + 130576T^4 + 364800*T^2 + 389376
$13$	\( (T^{6} - 4 T^{5} - 23 T^{4} + \cdots + 88)^{2} \) (T^6 - 4T^5 - 23T^4 + 80T^3 + 56T^2 - 216*T + 88)^2
$17$	\( T^{12} + 120 T^{10} + \cdots + 278784 \) T^12 + 120T^10 + 4712T^8 + 72864T^6 + 403600T^4 + 664320*T^2 + 278784
$19$	\( T^{12} + 84 T^{10} + \cdots + 576 \) T^12 + 84T^10 + 1673T^8 + 9096T^6 + 16624T^4 + 6528*T^2 + 576
$23$	\( T^{12} + 168 T^{10} + \cdots + 2985984 \) T^12 + 168T^10 + 9968T^8 + 241152T^6 + 1992256T^4 + 5004288*T^2 + 2985984
$29$	\( (T^{6} + 6 T^{5} + \cdots - 1152)^{2} \) (T^6 + 6T^5 - 55T^4 - 192T^3 + 1024T^2 + 192*T - 1152)^2
$31$	\( (T^{6} + 6 T^{5} + \cdots - 3488)^{2} \) (T^6 + 6T^5 - 91T^4 - 388T^3 + 2320T^2 + 4448*T - 3488)^2
$37$	\( T^{12} + \cdots + 160579584 \) T^12 + 248T^10 + 19512T^8 + 681056T^6 + 11197840T^4 + 79027200*T^2 + 160579584
$41$	\( T^{12} + 100 T^{10} + \cdots + 1296 \) T^12 + 100T^10 + 3153T^8 + 32632T^6 + 102280T^4 + 53280*T^2 + 1296
$43$	\( T^{12} + \cdots + 6321363049 \) T^12 - 20T^11 + 102T^10 + 532T^9 - 6249T^8 - 12520T^7 + 350388T^6 - 538360T^5 - 11554401T^4 + 42297724T^3 + 348717702T^2 - 2940168860*T + 6321363049
$47$	\( T^{12} + 152 T^{10} + \cdots + 63489024 \) T^12 + 152T^10 + 8832T^8 + 249728T^6 + 3598144T^4 + 24794112*T^2 + 63489024
$53$	\( T^{12} + \cdots + 160579584 \) T^12 + 436T^10 + 71076T^8 + 5278336T^6 + 168262144T^4 + 1490706432*T^2 + 160579584
$59$	\( T^{12} + \cdots + 375366979584 \) T^12 + 600T^10 + 139976T^8 + 16317792T^6 + 1003956112T^4 + 30956196864*T^2 + 375366979584
$61$	\( T^{12} + \cdots + 22647842064 \) T^12 + 500T^10 + 95745T^8 + 8785112T^6 + 389888200T^4 + 7067750304*T^2 + 22647842064
$67$	\( (T^{6} + 2 T^{5} + \cdots - 2288)^{2} \) (T^6 + 2T^5 - 55T^4 - 28T^3 + 704T^2 + 80*T - 2288)^2
$71$	\( (T^{6} - 252 T^{4} + \cdots - 228096)^{2} \) (T^6 - 252T^4 + 16128T^2 - 6912*T - 228096)^2
$73$	\( T^{12} + 284 T^{10} + \cdots + 627264 \) T^12 + 284T^10 + 20193T^8 + 525848T^6 + 4795696T^4 + 8673408*T^2 + 627264
$79$	\( (T^{6} + 4 T^{5} + \cdots - 388224)^{2} \) (T^6 + 4T^5 - 359T^4 - 816T^3 + 31968T^2 + 55872*T - 388224)^2
$83$	\( T^{12} + \cdots + 125619264 \) T^12 + 516T^10 + 94796T^8 + 7447584T^6 + 238932016T^4 + 2375040576*T^2 + 125619264
$89$	\( (T^{6} + 8 T^{5} + \cdots + 3168)^{2} \) (T^6 + 8T^5 - 108T^4 - 1192T^3 - 3368T^2 - 1632*T + 3168)^2
$97$	\( (T^{6} - 12 T^{5} + \cdots - 32768)^{2} \) (T^6 - 12T^5 - 232T^4 + 2176T^3 + 10240T^2 - 8192*T - 32768)^2
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\(n\)	\(1721\)	\(1981\)	\(3097\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)