LMFDB - Newform orbit 4056.2.c.q

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4056,2,Mod(337,4056)] 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("4056.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,-12,0,0,0,0,0,12,0,0,0,0,0,0,0,10,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$32.3873230598$
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} + 39x^{10} + 601x^{8} + 4599x^{6} + 17849x^{4} + 31203x^{2} + 16129 $ x^12 + 39x^10 + 601x^8 + 4599x^6 + 17849x^4 + 31203*x^2 + 16129
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
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^{3} - \beta_{10} q^{5} + (\beta_{9} - \beta_1) q^{7} + q^{9} + (\beta_{11} + \beta_{9} - \beta_{6} + \beta_1) q^{11} + \beta_{10} q^{15} + ( - \beta_{7} - \beta_{3} + 2 \beta_{2}) q^{17} + ( - \beta_{10} - 3 \beta_{9} + \cdots + \beta_{5}) q^{19}+ \cdots + (\beta_{11} + \beta_{9} - \beta_{6} + \beta_1) q^{99}+O(q^{100}) $ q - q^3 - b10 * q^5 + (b9 - b1) * q^7 + q^9 + (b11 + b9 - b6 + b1) * q^11 + b10 * q^15 + (-b7 - b3 + 2b2) q^17 + (-b10 - 3b9 + b6 + b5) q^19 + (-b9 + b1) * q^21 + (-b8 + b7 - b4 - 3b3) q^23 + (3b4 + 2b3 - b2 - 1) * q^25 - q^27 + (b7 - b4 - b3) * q^29 + (-b10 - 3b9 - 4b6 + b5) * q^31 + (-b11 - b9 + b6 - b1) * q^33 + (-b8 - b7 + 2b4 - 4b3) * q^35 + (b11 - b9 - 2b6 + 2b5 - 3b1) q^37 + (-2b10 + 3b9 + 2b6 + b5 - b1) q^41 + (b8 - b7 - 5b4 - 4b3 - b2 - 5) * q^43 - b10 * q^45 + (b10 - 3b9 - 5b6 + 5b5 - 2b1) * q^47 + (2b8 + b7 - b4 + b3 - 1) q^49 + (b7 + b3 - 2b2) q^51 + (2b4 + 5b3 + 2b2 + 4) q^53 + (3b8 - 2b7 - 7b4 - 3b3 - 3) * q^55 + (b10 + 3b9 - b6 - b5) q^57 + (2b11 + b10 + b9 + 5b6 - 2b5) q^59 + (b7 + 4b4 - b3 + 4) q^61 + (b9 - b1) * q^63 + (3b11 + 4b6 - 3b5 + 2b1) * q^67 + (b8 - b7 + b4 + 3b3) q^69 + (b11 - b10 - 9b9 - 7b6 + 6b5 - 2b1) * q^71 + (-2b11 + b10 - 8b9) * q^73 + (-3b4 - 2b3 + b2 + 1) * q^75 + (-2b7 + 9b4 + 5b3 + 3b2 + 8) * q^77 + (b7 - b4 - b3 - b2 + 2) * q^79 + q^81 + (b11 - b10 + 2b9 - b6 - 6b5 - b1) * q^83 + (-2b10 + 5b9 + 10b6 - 13b5 + b1) * q^85 + (-b7 + b4 + b3) * q^87 + (-b11 + 2b10 + 4b9 + 2b6 - 3b5 + 2b1) q^89 + (b10 + 3b9 + 4b6 - b5) * q^93 + (-b8 + 4b7 + 5b4 + 2b3 - 4) q^95 + (-b11 + 3b10 + 6b9 + 4b6 - 7b5 + 2b1) q^97 + (b11 + b9 - b6 + b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 12 q^{9} + 10 q^{17} + 12 q^{23} - 34 q^{25} - 12 q^{27} + 6 q^{29} + 8 q^{35} - 22 q^{43} - 10 q^{49} - 10 q^{51} + 24 q^{53} + 14 q^{55} + 34 q^{61} - 12 q^{69} + 34 q^{75} + 50 q^{77}+ \cdots - 86 q^{95}+O(q^{100}) $ 12 * q - 12 * q^3 + 12 * q^9 + 10 * q^17 + 12 * q^23 - 34 * q^25 - 12 * q^27 + 6 * q^29 + 8 * q^35 - 22 * q^43 - 10 * q^49 - 10 * q^51 + 24 * q^53 + 14 * q^55 + 34 * q^61 - 12 * q^69 + 34 * q^75 + 50 * q^77 + 28 * q^79 + 12 * q^81 - 6 * q^87 - 86 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 39x^{10} + 601x^{8} + 4599x^{6} + 17849x^{4} + 31203x^{2} + 16129 $ x^12 + 39*x^10 + 601*x^8 + 4599*x^6 + 17849*x^4 + 31203*x^2 + 16129 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 56\nu^{10} + 1762\nu^{8} + 20390\nu^{6} + 104189\nu^{4} + 216988\nu^{2} + 121298 ) / 167 $ (56v^10 + 1762v^8 + 20390v^6 + 104189v^4 + 216988*v^2 + 121298) / 167
$\beta_{3}$	$=$	$ ( 84\nu^{10} + 2643\nu^{8} + 30585\nu^{6} + 156200\nu^{4} + 324480\nu^{2} + 179776 ) / 167 $ (84v^10 + 2643v^8 + 30585v^6 + 156200v^4 + 324480*v^2 + 179776) / 167
$\beta_{4}$	$=$	$ ( 93\nu^{10} + 2956\nu^{8} + 34524\nu^{6} + 177731\nu^{4} + 371556\nu^{2} + 206672 ) / 167 $ (93v^10 + 2956v^8 + 34524v^6 + 177731v^4 + 371556*v^2 + 206672) / 167
$\beta_{5}$	$=$	$ ( 1680\nu^{11} + 52693\nu^{9} + 608360\nu^{7} + 3102123\nu^{5} + 6433154\nu^{3} + 3538740\nu ) / 21209 $ (1680v^11 + 52693v^9 + 608360v^7 + 3102123v^5 + 6433154v^3 + 3538740v) / 21209
$\beta_{6}$	$=$	$ ( 2097\nu^{11} + 66416\nu^{9} + 772998\nu^{7} + 3964790\nu^{5} + 8252119\nu^{3} + 4572386\nu ) / 21209 $ (2097v^11 + 66416v^9 + 772998v^7 + 3964790v^5 + 8252119v^3 + 4572386v) / 21209
$\beta_{7}$	$=$	$ ( 177\nu^{10} + 5599\nu^{8} + 65109\nu^{6} + 333931\nu^{4} + 696203\nu^{2} + 387617 ) / 167 $ (177v^10 + 5599v^8 + 65109v^6 + 333931v^4 + 696203*v^2 + 387617) / 167
$\beta_{8}$	$=$	$ ( 250\nu^{10} + 7878\nu^{8} + 91325\nu^{6} + 467378\nu^{4} + 973444\nu^{2} + 541330 ) / 167 $ (250v^10 + 7878v^8 + 91325v^6 + 467378v^4 + 973444*v^2 + 541330) / 167
$\beta_{9}$	$=$	$ ( 34\nu^{11} + 1072\nu^{9} + 12433\nu^{7} + 63656\nu^{5} + 132648\nu^{3} + 73858\nu ) / 127 $ (34v^11 + 1072v^9 + 12433v^7 + 63656v^5 + 132648v^3 + 73858v) / 127
$\beta_{10}$	$=$	$ ( 6670\nu^{11} + 209330\nu^{9} + 2416344\nu^{7} + 12308971\nu^{5} + 25489898\nu^{3} + 14036006\nu ) / 21209 $ (6670v^11 + 209330v^9 + 2416344v^7 + 12308971v^5 + 25489898v^3 + 14036006v) / 21209
$\beta_{11}$	$=$	$ ( 16384\nu^{11} + 518326\nu^{9} + 6027894\nu^{7} + 30916018\nu^{5} + 64425391\nu^{3} + 35710964\nu ) / 21209 $ (16384v^11 + 518326v^9 + 6027894v^7 + 30916018v^5 + 64425391v^3 + 35710964v) / 21209

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{7} - \beta_{4} - \beta_{3} - 7 $ b7 - b4 - b3 - 7
$\nu^{3}$	$=$	$ -\beta_{11} + 2\beta_{9} + 4\beta_{6} - 2\beta_{5} - 8\beta_1 $ -b11 + 2b9 + 4b6 - 2b5 - 8b1
$\nu^{4}$	$=$	$ -12\beta_{7} + 12\beta_{4} + 10\beta_{3} + 3\beta_{2} + 58 $ -12b7 + 12b4 + 10b3 + 3b2 + 58
$\nu^{5}$	$=$	$ 12\beta_{11} + \beta_{10} - 20\beta_{9} - 54\beta_{6} + 14\beta_{5} + 70\beta_1 $ 12b11 + b10 - 20b9 - 54b6 + 14b5 + 70*b1
$\nu^{6}$	$=$	$ 5\beta_{8} + 124\beta_{7} - 126\beta_{4} - 102\beta_{3} - 52\beta_{2} - 514 $ 5b8 + 124b7 - 126b4 - 102b3 - 52*b2 - 514
$\nu^{7}$	$=$	$ -126\beta_{11} - 28\beta_{10} + 191\beta_{9} + 598\beta_{6} - 52\beta_{5} - 638\beta_1 $ -126b11 - 28b10 + 191b9 + 598b6 - 52b5 - 638b1
$\nu^{8}$	$=$	$ -111\beta_{8} - 1236\beta_{7} + 1286\beta_{4} + 1063\beta_{3} + 672\beta_{2} + 4718 $ -111b8 - 1236b7 + 1286b4 + 1063b3 + 672*b2 + 4718
$\nu^{9}$	$=$	$ 1286\beta_{11} + 449\beta_{10} - 1856\beta_{9} - 6238\beta_{6} - 265\beta_{5} + 5954\beta_1 $ 1286b11 + 449b10 - 1856b9 - 6238b6 - 265b5 + 5954b1
$\nu^{10}$	$=$	$ 1672\beta_{8} + 12192\beta_{7} - 13037\beta_{4} - 11038\beta_{3} - 7789\beta_{2} - 44250 $ 1672b8 + 12192b7 - 13037b4 - 11038b3 - 7789*b2 - 44250
$\nu^{11}$	$=$	$ -13037\beta_{11} - 5790\beta_{10} + 18320\beta_{9} + 63501\beta_{6} + 8962\beta_{5} - 56442\beta_1 $ -13037b11 - 5790b10 + 18320b9 + 63501b6 + 8962b5 - 56442b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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} $

337.1

	−	0.920510i
		3.12925i
		1.88227i
	−	2.72245i
	−	3.16419i
	−	2.71914i
		2.71914i
		3.16419i
		2.72245i
	−	1.88227i
	−	3.12925i
		0.920510i

−1.00000

−

3.90568i

−

0.0794899i

1.00000

337.2

−1.00000

−

3.34715i

−

2.12925i

1.00000

337.3

−1.00000

−

2.90211i

−

2.88227i

1.00000

337.4

−1.00000

−

2.65870i

3.72245i

1.00000

337.5

−1.00000

−

2.21013i

4.16419i

1.00000

337.6

−1.00000

−

0.408195i

1.71914i

1.00000

337.7

−1.00000

0.408195i

−

1.71914i

1.00000

337.8

−1.00000

2.21013i

−

4.16419i

1.00000

337.9

−1.00000

2.65870i

−

3.72245i

1.00000

337.10

−1.00000

2.90211i

2.88227i

1.00000

337.11

−1.00000

3.34715i

2.12925i

1.00000

337.12

−1.00000

3.90568i

0.0794899i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4056.2.c.q		12
13.b	even	2	1	inner	4056.2.c.q		12
13.d	odd	4	1		4056.2.a.bf	✓	6
13.d	odd	4	1		4056.2.a.bg	yes	6
52.f	even	4	1		8112.2.a.cv		6
52.f	even	4	1		8112.2.a.cw		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4056.2.a.bf	✓	6	13.d	odd	4	1
4056.2.a.bg	yes	6	13.d	odd	4	1
4056.2.c.q		12	1.a	even	1	1	trivial
4056.2.c.q		12	13.b	even	2	1	inner
8112.2.a.cv		6	52.f	even	4	1
8112.2.a.cw		6	52.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4056, [\chi])$:

$ T_{5}^{12} + 47T_{5}^{10} + 853T_{5}^{8} + 7491T_{5}^{6} + 32025T_{5}^{4} + 54831T_{5}^{2} + 8281 $

T5^12 + 47*T5^10 + 853*T5^8 + 7491*T5^6 + 32025*T5^4 + 54831*T5^2 + 8281

$ T_{7}^{12} + 47T_{7}^{10} + 809T_{7}^{8} + 6271T_{7}^{6} + 21681T_{7}^{4} + 26883T_{7}^{2} + 169 $

T7^12 + 47*T7^10 + 809*T7^8 + 6271*T7^6 + 21681*T7^4 + 26883*T7^2 + 169

$ T_{11}^{12} + 110T_{11}^{10} + 4521T_{11}^{8} + 87550T_{11}^{6} + 829066T_{11}^{4} + 3517612T_{11}^{2} + 4618201 $

T11^12 + 110*T11^10 + 4521*T11^8 + 87550*T11^6 + 829066*T11^4 + 3517612*T11^2 + 4618201

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T + 1)^{12} $ (T + 1)^12
$5$	$ T^{12} + 47 T^{10} + \cdots + 8281 $ T^12 + 47T^10 + 853T^8 + 7491T^6 + 32025T^4 + 54831*T^2 + 8281
$7$	$ T^{12} + 47 T^{10} + \cdots + 169 $ T^12 + 47T^10 + 809T^8 + 6271T^6 + 21681T^4 + 26883*T^2 + 169
$11$	$ T^{12} + 110 T^{10} + \cdots + 4618201 $ T^12 + 110T^10 + 4521T^8 + 87550T^6 + 829066T^4 + 3517612*T^2 + 4618201
$13$	$ T^{12} $ T^12
$17$	$ (T^{6} - 5 T^{5} + \cdots - 26648)^{2} $ (T^6 - 5T^5 - 86T^4 + 329T^3 + 2544T^2 - 5396*T - 26648)^2
$19$	$ T^{12} + 121 T^{10} + \cdots + 4096 $ T^12 + 121T^10 + 4086T^8 + 26201T^6 + 51152T^4 + 29184*T^2 + 4096
$23$	$ (T^{6} - 6 T^{5} + \cdots + 1352)^{2} $ (T^6 - 6T^5 - 53T^4 + 357T^3 + 68T^2 - 2420*T + 1352)^2
$29$	$ (T^{6} - 3 T^{5} - 29 T^{4} + \cdots + 29)^{2} $ (T^6 - 3T^5 - 29T^4 + 21T^3 + 209T^2 + 151*T + 29)^2
$31$	$ T^{12} + 191 T^{10} + \cdots + 12453841 $ T^12 + 191T^10 + 13221T^8 + 399791T^6 + 4901277T^4 + 14562899*T^2 + 12453841
$37$	$ T^{12} + \cdots + 77404142656 $ T^12 + 438T^10 + 76397T^8 + 6785233T^6 + 323806216T^4 + 7901510096*T^2 + 77404142656
$41$	$ T^{12} + \cdots + 1080568384 $ T^12 + 285T^10 + 31142T^8 + 1646401T^6 + 42887236T^4 + 474860960*T^2 + 1080568384
$43$	$ (T^{6} + 11 T^{5} + \cdots + 5944)^{2} $ (T^6 + 11T^5 - 89T^4 - 1119T^3 - 1648T^2 + 4972*T + 5944)^2
$47$	$ T^{12} + \cdots + 1163628544 $ T^12 + 381T^10 + 52446T^8 + 3306057T^6 + 96367184T^4 + 1083087360*T^2 + 1163628544
$53$	$ (T^{6} - 12 T^{5} + \cdots + 30997)^{2} $ (T^6 - 12T^5 - 135T^4 + 2352T^3 - 8264T^2 - 3058*T + 30997)^2
$59$	$ T^{12} + 405 T^{10} + \cdots + 25969216 $ T^12 + 405T^10 + 60506T^8 + 4188409T^6 + 136391892T^4 + 1694090720*T^2 + 25969216
$61$	$ (T^{6} - 17 T^{5} + \cdots + 9304)^{2} $ (T^6 - 17T^5 + 6T^4 + 1085T^3 - 4222T^2 - 3048*T + 9304)^2
$67$	$ T^{12} + \cdots + 58474977856 $ T^12 + 545T^10 + 106310T^8 + 9665361T^6 + 428162596T^4 + 8540777440*T^2 + 58474977856
$71$	$ T^{12} + \cdots + 106291648576 $ T^12 + 654T^10 + 153505T^8 + 15807469T^6 + 737950980T^4 + 15143235168*T^2 + 106291648576
$73$	$ T^{12} + 607 T^{10} + \cdots + 47458321 $ T^12 + 607T^10 + 126221T^8 + 10081847T^6 + 229387349T^4 + 1553572835*T^2 + 47458321
$79$	$ (T^{6} - 14 T^{5} + \cdots + 4843)^{2} $ (T^6 - 14T^5 + 39T^4 + 210T^3 - 956T^2 - 686*T + 4843)^2
$83$	$ T^{12} + \cdots + 633869153281 $ T^12 + 647T^10 + 165105T^8 + 21219115T^6 + 1442483301T^4 + 48825662067*T^2 + 633869153281
$89$	$ T^{12} + \cdots + 247621696 $ T^12 + 369T^10 + 42034T^8 + 1507185T^6 + 21657524T^4 + 125394528*T^2 + 247621696
$97$	$ T^{12} + \cdots + 680322889 $ T^12 + 696T^10 + 154856T^8 + 13439411T^6 + 385356738T^4 + 1209107401*T^2 + 680322889
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - q^{3} - \beta_{10} q^{5} + (\beta_{9} - \beta_1) q^{7} + q^{9} + (\beta_{11} + \beta_{9} - \beta_{6} + \beta_1) q^{11} + \beta_{10} q^{15} + ( - \beta_{7} - \beta_{3} + 2 \beta_{2}) q^{17} + ( - \beta_{10} - 3 \beta_{9} + \cdots + \beta_{5}) q^{19}+ \cdots + (\beta_{11} + \beta_{9} - \beta_{6} + \beta_1) q^{99}+O(q^{100}) \) q - q^3 - b10 * q^5 + (b9 - b1) * q^7 + q^9 + (b11 + b9 - b6 + b1) * q^11 + b10 * q^15 + (-b7 - b3 + 2b2) q^17 + (-b10 - 3b9 + b6 + b5) q^19 + (-b9 + b1) * q^21 + (-b8 + b7 - b4 - 3b3) q^23 + (3b4 + 2b3 - b2 - 1) * q^25 - q^27 + (b7 - b4 - b3) * q^29 + (-b10 - 3b9 - 4b6 + b5) * q^31 + (-b11 - b9 + b6 - b1) * q^33 + (-b8 - b7 + 2b4 - 4b3) * q^35 + (b11 - b9 - 2b6 + 2b5 - 3b1) q^37 + (-2b10 + 3b9 + 2b6 + b5 - b1) q^41 + (b8 - b7 - 5b4 - 4b3 - b2 - 5) * q^43 - b10 * q^45 + (b10 - 3b9 - 5b6 + 5b5 - 2b1) * q^47 + (2b8 + b7 - b4 + b3 - 1) q^49 + (b7 + b3 - 2b2) q^51 + (2b4 + 5b3 + 2b2 + 4) q^53 + (3b8 - 2b7 - 7b4 - 3b3 - 3) * q^55 + (b10 + 3b9 - b6 - b5) q^57 + (2b11 + b10 + b9 + 5b6 - 2b5) q^59 + (b7 + 4b4 - b3 + 4) q^61 + (b9 - b1) * q^63 + (3b11 + 4b6 - 3b5 + 2b1) * q^67 + (b8 - b7 + b4 + 3b3) q^69 + (b11 - b10 - 9b9 - 7b6 + 6b5 - 2b1) * q^71 + (-2b11 + b10 - 8b9) * q^73 + (-3b4 - 2b3 + b2 + 1) * q^75 + (-2b7 + 9b4 + 5b3 + 3b2 + 8) * q^77 + (b7 - b4 - b3 - b2 + 2) * q^79 + q^81 + (b11 - b10 + 2b9 - b6 - 6b5 - b1) * q^83 + (-2b10 + 5b9 + 10b6 - 13b5 + b1) * q^85 + (-b7 + b4 + b3) * q^87 + (-b11 + 2b10 + 4b9 + 2b6 - 3b5 + 2b1) q^89 + (b10 + 3b9 + 4b6 - b5) * q^93 + (-b8 + 4b7 + 5b4 + 2b3 - 4) q^95 + (-b11 + 3b10 + 6b9 + 4b6 - 7b5 + 2b1) q^97 + (b11 + b9 - b6 + b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 12 q^{9} + 10 q^{17} + 12 q^{23} - 34 q^{25} - 12 q^{27} + 6 q^{29} + 8 q^{35} - 22 q^{43} - 10 q^{49} - 10 q^{51} + 24 q^{53} + 14 q^{55} + 34 q^{61} - 12 q^{69} + 34 q^{75} + 50 q^{77}+ \cdots - 86 q^{95}+O(q^{100}) \) 12 * q - 12 * q^3 + 12 * q^9 + 10 * q^17 + 12 * q^23 - 34 * q^25 - 12 * q^27 + 6 * q^29 + 8 * q^35 - 22 * q^43 - 10 * q^49 - 10 * q^51 + 24 * q^53 + 14 * q^55 + 34 * q^61 - 12 * q^69 + 34 * q^75 + 50 * q^77 + 28 * q^79 + 12 * q^81 - 6 * q^87 - 86 * q^95

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Newspace parameters

Newform invariants

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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	4056.2.c.q

Level	$4056$
Weight	$2$
Character orbit	4056.c
Analytic conductor	$32.387$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(32.3873230598\)
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:	\( x^{12} + 39x^{10} + 601x^{8} + 4599x^{6} + 17849x^{4} + 31203x^{2} + 16129 \) x^12 + 39x^10 + 601x^8 + 4599x^6 + 17849x^4 + 31203*x^2 + 16129
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 56\nu^{10} + 1762\nu^{8} + 20390\nu^{6} + 104189\nu^{4} + 216988\nu^{2} + 121298 ) / 167 \) (56v^10 + 1762v^8 + 20390v^6 + 104189v^4 + 216988*v^2 + 121298) / 167
\(\beta_{3}\)	\(=\)	\( ( 84\nu^{10} + 2643\nu^{8} + 30585\nu^{6} + 156200\nu^{4} + 324480\nu^{2} + 179776 ) / 167 \) (84v^10 + 2643v^8 + 30585v^6 + 156200v^4 + 324480*v^2 + 179776) / 167
\(\beta_{4}\)	\(=\)	\( ( 93\nu^{10} + 2956\nu^{8} + 34524\nu^{6} + 177731\nu^{4} + 371556\nu^{2} + 206672 ) / 167 \) (93v^10 + 2956v^8 + 34524v^6 + 177731v^4 + 371556*v^2 + 206672) / 167
\(\beta_{5}\)	\(=\)	\( ( 1680\nu^{11} + 52693\nu^{9} + 608360\nu^{7} + 3102123\nu^{5} + 6433154\nu^{3} + 3538740\nu ) / 21209 \) (1680v^11 + 52693v^9 + 608360v^7 + 3102123v^5 + 6433154v^3 + 3538740v) / 21209
\(\beta_{6}\)	\(=\)	\( ( 2097\nu^{11} + 66416\nu^{9} + 772998\nu^{7} + 3964790\nu^{5} + 8252119\nu^{3} + 4572386\nu ) / 21209 \) (2097v^11 + 66416v^9 + 772998v^7 + 3964790v^5 + 8252119v^3 + 4572386v) / 21209
\(\beta_{7}\)	\(=\)	\( ( 177\nu^{10} + 5599\nu^{8} + 65109\nu^{6} + 333931\nu^{4} + 696203\nu^{2} + 387617 ) / 167 \) (177v^10 + 5599v^8 + 65109v^6 + 333931v^4 + 696203*v^2 + 387617) / 167
\(\beta_{8}\)	\(=\)	\( ( 250\nu^{10} + 7878\nu^{8} + 91325\nu^{6} + 467378\nu^{4} + 973444\nu^{2} + 541330 ) / 167 \) (250v^10 + 7878v^8 + 91325v^6 + 467378v^4 + 973444*v^2 + 541330) / 167
\(\beta_{9}\)	\(=\)	\( ( 34\nu^{11} + 1072\nu^{9} + 12433\nu^{7} + 63656\nu^{5} + 132648\nu^{3} + 73858\nu ) / 127 \) (34v^11 + 1072v^9 + 12433v^7 + 63656v^5 + 132648v^3 + 73858v) / 127
\(\beta_{10}\)	\(=\)	\( ( 6670\nu^{11} + 209330\nu^{9} + 2416344\nu^{7} + 12308971\nu^{5} + 25489898\nu^{3} + 14036006\nu ) / 21209 \) (6670v^11 + 209330v^9 + 2416344v^7 + 12308971v^5 + 25489898v^3 + 14036006v) / 21209
\(\beta_{11}\)	\(=\)	\( ( 16384\nu^{11} + 518326\nu^{9} + 6027894\nu^{7} + 30916018\nu^{5} + 64425391\nu^{3} + 35710964\nu ) / 21209 \) (16384v^11 + 518326v^9 + 6027894v^7 + 30916018v^5 + 64425391v^3 + 35710964v) / 21209

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{7} - \beta_{4} - \beta_{3} - 7 \) b7 - b4 - b3 - 7
\(\nu^{3}\)	\(=\)	\( -\beta_{11} + 2\beta_{9} + 4\beta_{6} - 2\beta_{5} - 8\beta_1 \) -b11 + 2b9 + 4b6 - 2b5 - 8b1
\(\nu^{4}\)	\(=\)	\( -12\beta_{7} + 12\beta_{4} + 10\beta_{3} + 3\beta_{2} + 58 \) -12b7 + 12b4 + 10b3 + 3b2 + 58
\(\nu^{5}\)	\(=\)	\( 12\beta_{11} + \beta_{10} - 20\beta_{9} - 54\beta_{6} + 14\beta_{5} + 70\beta_1 \) 12b11 + b10 - 20b9 - 54b6 + 14b5 + 70*b1
\(\nu^{6}\)	\(=\)	\( 5\beta_{8} + 124\beta_{7} - 126\beta_{4} - 102\beta_{3} - 52\beta_{2} - 514 \) 5b8 + 124b7 - 126b4 - 102b3 - 52*b2 - 514
\(\nu^{7}\)	\(=\)	\( -126\beta_{11} - 28\beta_{10} + 191\beta_{9} + 598\beta_{6} - 52\beta_{5} - 638\beta_1 \) -126b11 - 28b10 + 191b9 + 598b6 - 52b5 - 638b1
\(\nu^{8}\)	\(=\)	\( -111\beta_{8} - 1236\beta_{7} + 1286\beta_{4} + 1063\beta_{3} + 672\beta_{2} + 4718 \) -111b8 - 1236b7 + 1286b4 + 1063b3 + 672*b2 + 4718
\(\nu^{9}\)	\(=\)	\( 1286\beta_{11} + 449\beta_{10} - 1856\beta_{9} - 6238\beta_{6} - 265\beta_{5} + 5954\beta_1 \) 1286b11 + 449b10 - 1856b9 - 6238b6 - 265b5 + 5954b1
\(\nu^{10}\)	\(=\)	\( 1672\beta_{8} + 12192\beta_{7} - 13037\beta_{4} - 11038\beta_{3} - 7789\beta_{2} - 44250 \) 1672b8 + 12192b7 - 13037b4 - 11038b3 - 7789*b2 - 44250
\(\nu^{11}\)	\(=\)	\( -13037\beta_{11} - 5790\beta_{10} + 18320\beta_{9} + 63501\beta_{6} + 8962\beta_{5} - 56442\beta_1 \) -13037b11 - 5790b10 + 18320b9 + 63501b6 + 8962b5 - 56442b1

\(n\)	\(1015\)	\(2029\)	\(2705\)	\(3889\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 337.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T + 1)^{12} \) (T + 1)^12
$5$	\( T^{12} + 47 T^{10} + \cdots + 8281 \) T^12 + 47T^10 + 853T^8 + 7491T^6 + 32025T^4 + 54831*T^2 + 8281
$7$	\( T^{12} + 47 T^{10} + \cdots + 169 \) T^12 + 47T^10 + 809T^8 + 6271T^6 + 21681T^4 + 26883*T^2 + 169
$11$	\( T^{12} + 110 T^{10} + \cdots + 4618201 \) T^12 + 110T^10 + 4521T^8 + 87550T^6 + 829066T^4 + 3517612*T^2 + 4618201
$13$	\( T^{12} \) T^12
$17$	\( (T^{6} - 5 T^{5} + \cdots - 26648)^{2} \) (T^6 - 5T^5 - 86T^4 + 329T^3 + 2544T^2 - 5396*T - 26648)^2
$19$	\( T^{12} + 121 T^{10} + \cdots + 4096 \) T^12 + 121T^10 + 4086T^8 + 26201T^6 + 51152T^4 + 29184*T^2 + 4096
$23$	\( (T^{6} - 6 T^{5} + \cdots + 1352)^{2} \) (T^6 - 6T^5 - 53T^4 + 357T^3 + 68T^2 - 2420*T + 1352)^2
$29$	\( (T^{6} - 3 T^{5} - 29 T^{4} + \cdots + 29)^{2} \) (T^6 - 3T^5 - 29T^4 + 21T^3 + 209T^2 + 151*T + 29)^2
$31$	\( T^{12} + 191 T^{10} + \cdots + 12453841 \) T^12 + 191T^10 + 13221T^8 + 399791T^6 + 4901277T^4 + 14562899*T^2 + 12453841
$37$	\( T^{12} + \cdots + 77404142656 \) T^12 + 438T^10 + 76397T^8 + 6785233T^6 + 323806216T^4 + 7901510096*T^2 + 77404142656
$41$	\( T^{12} + \cdots + 1080568384 \) T^12 + 285T^10 + 31142T^8 + 1646401T^6 + 42887236T^4 + 474860960*T^2 + 1080568384
$43$	\( (T^{6} + 11 T^{5} + \cdots + 5944)^{2} \) (T^6 + 11T^5 - 89T^4 - 1119T^3 - 1648T^2 + 4972*T + 5944)^2
$47$	\( T^{12} + \cdots + 1163628544 \) T^12 + 381T^10 + 52446T^8 + 3306057T^6 + 96367184T^4 + 1083087360*T^2 + 1163628544
$53$	\( (T^{6} - 12 T^{5} + \cdots + 30997)^{2} \) (T^6 - 12T^5 - 135T^4 + 2352T^3 - 8264T^2 - 3058*T + 30997)^2
$59$	\( T^{12} + 405 T^{10} + \cdots + 25969216 \) T^12 + 405T^10 + 60506T^8 + 4188409T^6 + 136391892T^4 + 1694090720*T^2 + 25969216
$61$	\( (T^{6} - 17 T^{5} + \cdots + 9304)^{2} \) (T^6 - 17T^5 + 6T^4 + 1085T^3 - 4222T^2 - 3048*T + 9304)^2
$67$	\( T^{12} + \cdots + 58474977856 \) T^12 + 545T^10 + 106310T^8 + 9665361T^6 + 428162596T^4 + 8540777440*T^2 + 58474977856
$71$	\( T^{12} + \cdots + 106291648576 \) T^12 + 654T^10 + 153505T^8 + 15807469T^6 + 737950980T^4 + 15143235168*T^2 + 106291648576
$73$	\( T^{12} + 607 T^{10} + \cdots + 47458321 \) T^12 + 607T^10 + 126221T^8 + 10081847T^6 + 229387349T^4 + 1553572835*T^2 + 47458321
$79$	\( (T^{6} - 14 T^{5} + \cdots + 4843)^{2} \) (T^6 - 14T^5 + 39T^4 + 210T^3 - 956T^2 - 686*T + 4843)^2
$83$	\( T^{12} + \cdots + 633869153281 \) T^12 + 647T^10 + 165105T^8 + 21219115T^6 + 1442483301T^4 + 48825662067*T^2 + 633869153281
$89$	\( T^{12} + \cdots + 247621696 \) T^12 + 369T^10 + 42034T^8 + 1507185T^6 + 21657524T^4 + 125394528*T^2 + 247621696
$97$	\( T^{12} + \cdots + 680322889 \) T^12 + 696T^10 + 154856T^8 + 13439411T^6 + 385356738T^4 + 1209107401*T^2 + 680322889
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