LMFDB - Newform orbit 896.2.t.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(65,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("896.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.t (of order $6$, degree $2$, 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:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 98x^{12} - 608x^{10} + 2427x^{8} + 1888x^{6} - 366x^{4} + 700x^{2} + 625 $ x^16 - 12x^14 + 98x^12 - 608x^10 + 2427x^8 + 1888x^6 - 366x^4 + 700*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{15} q^{3} - \beta_{2} q^{5} + (\beta_{14} - \beta_{11} + \beta_{10}) q^{7} + ( - \beta_{9} + \beta_{5} + 1) q^{9}+O(q^{10}) $ q - b15 * q^3 - b2 * q^5 + (b14 - b11 + b10) * q^7 + (-b9 + b5 + 1) * q^9	$ q - \beta_{15} q^{3} - \beta_{2} q^{5} + (\beta_{14} - \beta_{11} + \beta_{10}) q^{7} + ( - \beta_{9} + \beta_{5} + 1) q^{9} + ( - \beta_{15} + \beta_{13}) q^{11} + ( - \beta_{8} + \beta_{6}) q^{13} + ( - \beta_{14} + \beta_{11}) q^{15} + ( - \beta_{9} - 2 \beta_{5} - \beta_1) q^{17} + ( - \beta_{15} - \beta_{13} + \cdots - \beta_{3}) q^{19}+ \cdots + ( - 6 \beta_{7} + \beta_{3}) q^{99}+O(q^{100}) $ q - b15 * q^3 - b2 * q^5 + (b14 - b11 + b10) * q^7 + (-b9 + b5 + 1) * q^9 + (-b15 + b13) * q^11 + (-b8 + b6) * q^13 + (-b14 + b11) * q^15 + (-b9 - 2b5 - b1) q^17 + (-b15 - b13 + b7 - b3) * q^19 + (2b8 - b6 + b4 + b2) q^21 + (b11 - 3b10) q^23 + (-2b9 - 2b5 - 2b1) q^25 + (-b7 - b3) * q^27 + (-b8 + b6 - 2b4) q^29 + (-3b14 + 2b12) * q^31 + (-2b9 + 3b5 + 3) * q^33 + (b15 + b13 - b7 + 2b3) q^35 + (-3b6 + b2) q^37 + (2b14 - 5b12) * q^39 + (b1 + 5) * q^41 + (2b7 + 2b3) * q^43 + (-b8 + 2b4 + 2b2) * q^45 + (b11 + 4b10) q^47 + (-2b9 - b1 + 4) q^49 + (b15 - b13 - b7 - b3) * q^51 + (3b8 - b4 - b2) q^53 + (3b14 + 5b12 - 3b11 - 5b10) * q^55 + 5 * q^57 + (3b15 - 2b13) * q^59 + (b6 + b2) * q^61 + (-2b14 + 2b12 + 5b10) q^63 + (-b9 - b5 - 1) * q^65 + (5b15 + 2b13) * q^67 + (-4b8 + 4b6 - b4) * q^69 + (2b14 - 2b12 - 2b11 + 2b10) * q^71 + (-2b9 + 3b5 - 2b1) q^73 + (4b15 - 2b13 - 4b7 - 2b3) * q^75 + (3b8 - b6 - b4 - 3b2) * q^77 + (b11 - b10) * q^79 + (b9 + b1) * q^81 + 4b7 q^83 + (-b8 + b6 + 5b4) q^85 + (4b14 - 5b12) * q^87 + (-7b5 - 7) q^89 + (-2b15 + 4b7 + b3) * q^91 + (-5b6 - 3b2) * q^93 + (-5b14 - 5b12) * q^95 + (-3b1 - 1) q^97 + (-6b7 + b3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{9}+O(q^{10}) $ 16 * q + 8 * q^9	$ 16 q + 8 q^{9} + 16 q^{17} + 16 q^{25} + 24 q^{33} + 80 q^{41} + 64 q^{49} + 80 q^{57} - 8 q^{65} - 24 q^{73} - 56 q^{89} - 16 q^{97}+O(q^{100}) $ 16 * q + 8 * q^9 + 16 * q^17 + 16 * q^25 + 24 * q^33 + 80 * q^41 + 64 * q^49 + 80 * q^57 - 8 * q^65 - 24 * q^73 - 56 * q^89 - 16 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 12x^{14} + 98x^{12} - 608x^{10} + 2427x^{8} + 1888x^{6} - 366x^{4} + 700x^{2} + 625 $ x^16 - 12*x^14 + 98*x^12 - 608*x^10 + 2427*x^8 + 1888*x^6 - 366*x^4 + 700*x^2 + 625 :

$\beta_{1}$	$=$	$ ( 46 \nu^{14} + 10984 \nu^{12} + 4869 \nu^{10} - 31012 \nu^{8} - 14975 \nu^{6} + 3692 \nu^{4} + \cdots + 887080822 ) / 267466838 $ (46v^14 + 10984v^12 + 4869v^10 - 31012v^8 - 14975v^6 + 3692v^4 - 9025*v^2 + 887080822) / 267466838
$\beta_{2}$	$=$	$ ( 88901573 \nu^{14} - 1131343676 \nu^{12} + 9539640304 \nu^{10} - 61096184484 \nu^{8} + \cdots + 90100085000 ) / 46806696650 $ (88901573v^14 - 1131343676v^12 + 9539640304v^10 - 61096184484v^8 + 260603918896v^6 - 24479017876v^4 + 8929240657*v^2 + 90100085000) / 46806696650
$\beta_{3}$	$=$	$ ( - 100243982 \nu^{15} + 1310942259 \nu^{13} - 11054039436 \nu^{11} + 70587360231 \nu^{9} + \cdots - 104403296250 \nu ) / 234033483250 $ (-100243982v^15 + 1310942259v^13 - 11054039436v^11 + 70587360231v^9 - 301974278364v^7 + 28365013809v^5 + 386287043712v^3 - 104403296250v) / 234033483250
$\beta_{4}$	$=$	$ ( - 8511334 \nu^{14} + 107376168 \nu^{12} - 900906201 \nu^{10} + 5738119348 \nu^{8} + \cdots - 7594410356 ) / 1872267866 $ (-8511334v^14 + 107376168v^12 - 900906201v^10 + 5738119348v^8 - 24243341363v^6 - 683127068v^4 + 1669886725*v^2 - 7594410356) / 1872267866
$\beta_{5}$	$=$	$ ( 7370156 \nu^{14} - 94041222 \nu^{12} + 792968088 \nu^{10} - 5073733873 \nu^{8} + \cdots + 6257677325 ) / 1231755175 $ (7370156v^14 - 94041222v^12 + 792968088v^10 - 5073733873v^8 + 21662304312v^6 - 2034781122v^4 - 3146885796*v^2 + 6257677325) / 1231755175
$\beta_{6}$	$=$	$ ( 562150656 \nu^{14} - 7147069697 \nu^{12} + 60265041988 \nu^{10} - 385625932323 \nu^{8} + \cdots + 569191838750 ) / 46806696650 $ (562150656v^14 - 7147069697v^12 + 60265041988v^10 - 385625932323v^8 + 1646320575412v^6 - 154641998347v^4 + 56490278304*v^2 + 569191838750) / 46806696650
$\beta_{7}$	$=$	$ ( - 634901829 \nu^{15} + 8278918773 \nu^{13} - 69808943892 \nu^{11} + 446544853007 \nu^{9} + \cdots - 659332173750 \nu ) / 468066966500 $ (-634901829v^15 + 8278918773v^13 - 69808943892v^11 + 446544853007v^9 - 1907040912708v^7 + 179131951623v^5 + 2440229361389v^3 - 659332173750v) / 468066966500
$\beta_{8}$	$=$	$ ( - 781044744 \nu^{14} + 9798111678 \nu^{12} - 81909271112 \nu^{10} + 519921366477 \nu^{8} + \cdots - 630227236425 ) / 46806696650 $ (-781044744v^14 + 9798111678v^12 - 81909271112v^10 + 519921366477v^8 - 2182506988188v^6 - 262448029147v^4 + 320019375804*v^2 - 630227236425) / 46806696650
$\beta_{9}$	$=$	$ ( - 929099357 \nu^{14} + 11852042984 \nu^{12} - 99938002336 \nu^{10} + 639501846006 \nu^{8} + \cdots - 943895390000 ) / 46806696650 $ (-929099357v^14 + 11852042984v^12 - 99938002336v^10 + 639501846006v^8 - 2730106610464v^6 + 256444065784v^4 + 396622436737*v^2 - 943895390000) / 46806696650
$\beta_{10}$	$=$	$ ( 1144428498 \nu^{15} - 14797058726 \nu^{13} + 125576013804 \nu^{11} - 808425801909 \nu^{9} + \cdots + 1009835789225 \nu ) / 234033483250 $ (1144428498v^15 - 14797058726v^13 + 125576013804v^11 - 808425801909v^9 + 3494792883146v^7 - 869736666151v^5 - 504251713768v^3 + 1009835789225v) / 234033483250
$\beta_{11}$	$=$	$ ( - 2652364739 \nu^{15} + 34288646618 \nu^{13} - 290979178297 \nu^{11} + \cdots - 2339349545425 \nu ) / 468066966500 $ (-2652364739v^15 + 34288646618v^13 - 290979178297v^11 + 1873051965937v^9 - 8095938906053v^7 + 2014815612393v^5 + 1168228883974v^3 - 2339349545425v) / 468066966500
$\beta_{12}$	$=$	$ ( - 62239472 \nu^{15} + 785106535 \nu^{13} - 6587912808 \nu^{11} + 41960228384 \nu^{9} + \cdots - 69269810581 \nu ) / 9361339330 $ (-62239472v^15 + 785106535v^13 - 6587912808v^11 + 41960228384v^9 - 177396339082v^7 - 4995394144v^5 + 12211113800v^3 - 69269810581v) / 9361339330
$\beta_{13}$	$=$	$ ( - 62239472 \nu^{15} + 785106535 \nu^{13} - 6587912808 \nu^{11} + 41960228384 \nu^{9} + \cdots - 50547131921 \nu ) / 9361339330 $ (-62239472v^15 + 785106535v^13 - 6587912808v^11 + 41960228384v^9 - 177396339082v^7 - 4995394144v^5 + 12211113800v^3 - 50547131921v) / 9361339330
$\beta_{14}$	$=$	$ ( 144161746 \nu^{15} - 1818438765 \nu^{13} + 15259207419 \nu^{11} - 97190088412 \nu^{9} + \cdots + 160476040633 \nu ) / 18722678660 $ (144161746v^15 - 1818438765v^13 + 15259207419v^11 - 97190088412v^9 + 410972310431v^7 + 11570547092v^5 - 28283907775v^3 + 160476040633v) / 18722678660
$\beta_{15}$	$=$	$ ( 393116414 \nu^{15} - 4959633785 \nu^{13} + 41610517821 \nu^{11} - 265028831108 \nu^{9} + \cdots + 319291589437 \nu ) / 18722678660 $ (393116414v^15 - 4959633785v^13 + 41610517821v^11 - 265028831108v^9 + 1120558715009v^7 + 31551865228v^5 - 77127731225v^3 + 319291589437v) / 18722678660

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

$\nu$	$=$	$ ( \beta_{13} - \beta_{12} ) / 2 $ (b13 - b12) / 2
$\nu^{2}$	$=$	$ \beta_{9} + 3\beta_{5} + \beta_{2} + 3 $ b9 + 3*b5 + b2 + 3
$\nu^{3}$	$=$	$ ( -6\beta_{14} - 7\beta_{12} + 6\beta_{11} + 7\beta_{10} + 2\beta_{7} - 5\beta_{3} ) / 2 $ (-6b14 - 7b12 + 6b11 + 7b10 + 2b7 - 5b3) / 2
$\nu^{4}$	$=$	$ 4\beta_{9} - 2\beta_{8} + 13\beta_{5} + 12\beta_{4} + 12\beta_{2} + 4\beta_1 $ 4b9 - 2b8 + 13b5 + 12b4 + 12b2 + 4b1
$\nu^{5}$	$=$	$ ( -4\beta_{15} - 13\beta_{13} + 60\beta_{11} + 69\beta_{10} + 4\beta_{7} - 13\beta_{3} ) / 2 $ (-4b15 - 13b13 + 60b11 + 69b10 + 4b7 - 13b3) / 2
$\nu^{6}$	$=$	$ -16\beta_{8} + 16\beta_{6} + 101\beta_{4} - \beta _1 + 3 $ -16b8 + 16b6 + 101*b4 - b1 + 3
$\nu^{7}$	$=$	$ ( 34\beta_{15} + 434\beta_{14} + 107\beta_{13} + 503\beta_{12} ) / 2 $ (34b15 + 434b14 + 107b13 + 503b12) / 2
$\nu^{8}$	$=$	$ 224\beta_{9} + 100\beta_{6} + 743\beta_{5} - 632\beta_{2} + 743 $ 224b9 + 100b6 + 743b5 - 632b2 + 743
$\nu^{9}$	$=$	$ ( 2280\beta_{14} + 2641\beta_{12} - 2280\beta_{11} - 2641\beta_{10} + 648\beta_{7} - 2047\beta_{3} ) / 2 $ (2280b14 + 2641b12 - 2280b11 - 2641b10 + 648b7 - 2047b3) / 2
$\nu^{10}$	$=$	$ 2779\beta_{9} + 408\beta_{8} + 9217\beta_{5} - 2577\beta_{4} - 2577\beta_{2} + 2779\beta_1 $ 2779b9 + 408b8 + 9217b5 - 2577b4 - 2577b2 + 2779b1
$\nu^{11}$	$=$	$ ( -6374\beta_{15} - 20131\beta_{13} - 5566\beta_{11} - 6447\beta_{10} + 6374\beta_{7} - 20131\beta_{3} ) / 2 $ (-6374b15 - 20131b13 - 5566b11 - 6447b10 + 6374b7 - 20131b3) / 2
$\nu^{12}$	$=$	$ -202\beta_{8} + 202\beta_{6} + 1276\beta_{4} + 23116\beta _1 - 76667 $ -202b8 + 202b6 + 1276b4 + 23116b1 - 76667
$\nu^{13}$	$=$	$ ( -45828\beta_{15} + 51740\beta_{14} - 144739\beta_{13} + 59931\beta_{12} ) / 2 $ (-45828b15 + 51740b14 - 144739b13 + 59931b12) / 2
$\nu^{14}$	$=$	$ -143261\beta_{9} + 24392\beta_{6} - 475143\beta_{5} - 154075\beta_{2} - 475143 $ -143261b9 + 24392b6 - 475143b5 - 154075b2 - 475143
$\nu^{15}$	$=$	$ ( 951606 \beta_{14} + 1102257 \beta_{12} - 951606 \beta_{11} - 1102257 \beta_{10} + \cdots + 750851 \beta_{3} ) / 2 $ (951606b14 + 1102257b12 - 951606b11 - 1102257b10 - 237738b7 + 750851b3) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$1$	$-1 - \beta_{5}$	$-1$

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} $

65.1

−0.724406	−	0.398261i
−0.0172989	+	0.826484i
2.61463	−	0.693061i
1.90752	−	1.91781i
−2.61463	+	0.693061i
−1.90752	+	1.91781i
0.724406	+	0.398261i
0.0172989	−	0.826484i
−0.724406	+	0.398261i
−0.0172989	−	0.826484i
2.61463	+	0.693061i
1.90752	+	1.91781i
−2.61463	−	0.693061i
−1.90752	−	1.91781i
0.724406	−	0.398261i
0.0172989	+	0.826484i

−2.34253

1.35246i

−0.524464

−

0.302800i

−2.34521

1.22474i

2.15831

−

3.73831i

65.2

−2.34253

1.35246i

0.524464

0.302800i

2.34521

−

1.22474i

2.15831

−

3.73831i

65.3

−0.715913

0.413333i

−3.19765

−

1.84616i

−2.34521

−

1.22474i

−1.15831

2.00626i

65.4

−0.715913

0.413333i

3.19765

1.84616i

2.34521

1.22474i

−1.15831

2.00626i

65.5

0.715913

−

0.413333i

−3.19765

−

1.84616i

2.34521

1.22474i

−1.15831

2.00626i

65.6

0.715913

−

0.413333i

3.19765

1.84616i

−2.34521

−

1.22474i

−1.15831

2.00626i

65.7

2.34253

−

1.35246i

−0.524464

−

0.302800i

2.34521

−

1.22474i

2.15831

−

3.73831i

65.8

2.34253

−

1.35246i

0.524464

0.302800i

−2.34521

1.22474i

2.15831

−

3.73831i

193.1

−2.34253

−

1.35246i

−0.524464

0.302800i

−2.34521

−

1.22474i

2.15831

3.73831i

193.2

−2.34253

−

1.35246i

0.524464

−

0.302800i

2.34521

1.22474i

2.15831

3.73831i

193.3

−0.715913

−

0.413333i

−3.19765

1.84616i

−2.34521

1.22474i

−1.15831

−

2.00626i

193.4

−0.715913

−

0.413333i

3.19765

−

1.84616i

2.34521

−

1.22474i

−1.15831

−

2.00626i

193.5

0.715913

0.413333i

−3.19765

1.84616i

2.34521

−

1.22474i

−1.15831

−

2.00626i

193.6

0.715913

0.413333i

3.19765

−

1.84616i

−2.34521

1.22474i

−1.15831

−

2.00626i

193.7

2.34253

1.35246i

−0.524464

0.302800i

2.34521

1.22474i

2.15831

3.73831i

193.8

2.34253

1.35246i

0.524464

−

0.302800i

−2.34521

−

1.22474i

2.15831

3.73831i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.c	even	3	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
28.g	odd	6	1	inner
56.k	odd	6	1	inner
56.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	896.2.t.c	✓	16
4.b	odd	2	1	inner	896.2.t.c	✓	16
7.c	even	3	1	inner	896.2.t.c	✓	16
8.b	even	2	1	inner	896.2.t.c	✓	16
8.d	odd	2	1	inner	896.2.t.c	✓	16
28.g	odd	6	1	inner	896.2.t.c	✓	16
56.k	odd	6	1	inner	896.2.t.c	✓	16
56.p	even	6	1	inner	896.2.t.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
896.2.t.c	✓	16	1.a	even	1	1	trivial
896.2.t.c	✓	16	4.b	odd	2	1	inner
896.2.t.c	✓	16	7.c	even	3	1	inner
896.2.t.c	✓	16	8.b	even	2	1	inner
896.2.t.c	✓	16	8.d	odd	2	1	inner
896.2.t.c	✓	16	28.g	odd	6	1	inner
896.2.t.c	✓	16	56.k	odd	6	1	inner
896.2.t.c	✓	16	56.p	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 8T_{3}^{6} + 59T_{3}^{4} - 40T_{3}^{2} + 25 $ T3^8 - 8*T3^6 + 59*T3^4 - 40*T3^2 + 25 acting on $S_{2}^{\mathrm{new}}(896, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 8 T^{6} + 59 T^{4} + \cdots + 25)^{2} $ (T^8 - 8T^6 + 59T^4 - 40*T^2 + 25)^2
$5$	$ (T^{8} - 14 T^{6} + \cdots + 25)^{2} $ (T^8 - 14T^6 + 191T^4 - 70*T^2 + 25)^2
$7$	$ (T^{4} - 8 T^{2} + 49)^{4} $ (T^4 - 8*T^2 + 49)^4
$11$	$ (T^{8} - 32 T^{6} + \cdots + 60025)^{2} $ (T^8 - 32T^6 + 779T^4 - 7840*T^2 + 60025)^2
$13$	$ (T^{4} + 16 T^{2} + 20)^{4} $ (T^4 + 16*T^2 + 20)^4
$17$	$ (T^{4} - 4 T^{3} + 23 T^{2} + \cdots + 49)^{4} $ (T^4 - 4T^3 + 23T^2 + 28*T + 49)^4
$19$	$ (T^{8} - 40 T^{6} + \cdots + 15625)^{2} $ (T^8 - 40T^6 + 1475T^4 - 5000*T^2 + 15625)^2
$23$	$ (T^{8} + 36 T^{6} + \cdots + 2401)^{2} $ (T^8 + 36T^6 + 1247T^4 + 1764*T^2 + 2401)^2
$29$	$ (T^{4} + 64 T^{2} + 980)^{4} $ (T^4 + 64*T^2 + 980)^4
$31$	$ (T^{8} + 100 T^{6} + \cdots + 5764801)^{2} $ (T^8 + 100T^6 + 7599T^4 + 240100*T^2 + 5764801)^2
$37$	$ (T^{8} - 170 T^{6} + \cdots + 37515625)^{2} $ (T^8 - 170T^6 + 22775T^4 - 1041250*T^2 + 37515625)^2
$41$	$ (T^{2} - 10 T + 14)^{8} $ (T^2 - 10*T + 14)^8
$43$	$ (T^{4} + 128 T^{2} + 3920)^{4} $ (T^4 + 128*T^2 + 3920)^4
$47$	$ (T^{8} + 92 T^{6} + \cdots + 1500625)^{2} $ (T^8 + 92T^6 + 7239T^4 + 112700*T^2 + 1500625)^2
$53$	$ (T^{8} - 170 T^{6} + \cdots + 37515625)^{2} $ (T^8 - 170T^6 + 22775T^4 - 1041250*T^2 + 37515625)^2
$59$	$ (T^{8} - 160 T^{6} + \cdots + 37515625)^{2} $ (T^8 - 160T^6 + 19475T^4 - 980000*T^2 + 37515625)^2
$61$	$ (T^{8} - 26 T^{6} + \cdots + 15625)^{2} $ (T^8 - 26T^6 + 551T^4 - 3250*T^2 + 15625)^2
$67$	$ (T^{8} - 352 T^{6} + \cdots + 878826025)^{2} $ (T^8 - 352T^6 + 94259T^4 - 10435040*T^2 + 878826025)^2
$71$	$ (T^{4} - 48 T^{2} + 400)^{4} $ (T^4 - 48*T^2 + 400)^4
$73$	$ (T^{4} + 6 T^{3} + \cdots + 1225)^{4} $ (T^4 + 6T^3 + 71T^2 - 210*T + 1225)^4
$79$	$ (T^{8} + 12 T^{6} + \cdots + 625)^{2} $ (T^8 + 12T^6 + 119T^4 + 300*T^2 + 625)^2
$83$	$ (T^{4} + 128 T^{2} + 1280)^{4} $ (T^4 + 128*T^2 + 1280)^4
$89$	$ (T^{2} + 7 T + 49)^{8} $ (T^2 + 7*T + 49)^8
$97$	$ (T^{2} + 2 T - 98)^{8} $ (T^2 + 2*T - 98)^8
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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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{15} q^{3} - \beta_{2} q^{5} + (\beta_{14} - \beta_{11} + \beta_{10}) q^{7} + ( - \beta_{9} + \beta_{5} + 1) q^{9}+O(q^{10}) \) q - b15 * q^3 - b2 * q^5 + (b14 - b11 + b10) * q^7 + (-b9 + b5 + 1) * q^9	\( q - \beta_{15} q^{3} - \beta_{2} q^{5} + (\beta_{14} - \beta_{11} + \beta_{10}) q^{7} + ( - \beta_{9} + \beta_{5} + 1) q^{9} + ( - \beta_{15} + \beta_{13}) q^{11} + ( - \beta_{8} + \beta_{6}) q^{13} + ( - \beta_{14} + \beta_{11}) q^{15} + ( - \beta_{9} - 2 \beta_{5} - \beta_1) q^{17} + ( - \beta_{15} - \beta_{13} + \cdots - \beta_{3}) q^{19}+ \cdots + ( - 6 \beta_{7} + \beta_{3}) q^{99}+O(q^{100}) \) q - b15 * q^3 - b2 * q^5 + (b14 - b11 + b10) * q^7 + (-b9 + b5 + 1) * q^9 + (-b15 + b13) * q^11 + (-b8 + b6) * q^13 + (-b14 + b11) * q^15 + (-b9 - 2b5 - b1) q^17 + (-b15 - b13 + b7 - b3) * q^19 + (2b8 - b6 + b4 + b2) q^21 + (b11 - 3b10) q^23 + (-2b9 - 2b5 - 2b1) q^25 + (-b7 - b3) * q^27 + (-b8 + b6 - 2b4) q^29 + (-3b14 + 2b12) * q^31 + (-2b9 + 3b5 + 3) * q^33 + (b15 + b13 - b7 + 2b3) q^35 + (-3b6 + b2) q^37 + (2b14 - 5b12) * q^39 + (b1 + 5) * q^41 + (2b7 + 2b3) * q^43 + (-b8 + 2b4 + 2b2) * q^45 + (b11 + 4b10) q^47 + (-2b9 - b1 + 4) q^49 + (b15 - b13 - b7 - b3) * q^51 + (3b8 - b4 - b2) q^53 + (3b14 + 5b12 - 3b11 - 5b10) * q^55 + 5 * q^57 + (3b15 - 2b13) * q^59 + (b6 + b2) * q^61 + (-2b14 + 2b12 + 5b10) q^63 + (-b9 - b5 - 1) * q^65 + (5b15 + 2b13) * q^67 + (-4b8 + 4b6 - b4) * q^69 + (2b14 - 2b12 - 2b11 + 2b10) * q^71 + (-2b9 + 3b5 - 2b1) q^73 + (4b15 - 2b13 - 4b7 - 2b3) * q^75 + (3b8 - b6 - b4 - 3b2) * q^77 + (b11 - b10) * q^79 + (b9 + b1) * q^81 + 4b7 q^83 + (-b8 + b6 + 5b4) q^85 + (4b14 - 5b12) * q^87 + (-7b5 - 7) q^89 + (-2b15 + 4b7 + b3) * q^91 + (-5b6 - 3b2) * q^93 + (-5b14 - 5b12) * q^95 + (-3b1 - 1) q^97 + (-6b7 + b3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9}+O(q^{10}) \) 16 * q + 8 * q^9	\( 16 q + 8 q^{9} + 16 q^{17} + 16 q^{25} + 24 q^{33} + 80 q^{41} + 64 q^{49} + 80 q^{57} - 8 q^{65} - 24 q^{73} - 56 q^{89} - 16 q^{97}+O(q^{100}) \) 16 * q + 8 * q^9 + 16 * q^17 + 16 * q^25 + 24 * q^33 + 80 * q^41 + 64 * q^49 + 80 * q^57 - 8 * q^65 - 24 * q^73 - 56 * q^89 - 16 * q^97

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	896.2.t.c

Level	$896$
Weight	$2$
Character orbit	896.t
Analytic conductor	$7.155$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(7.15459602111\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 12x^{14} + 98x^{12} - 608x^{10} + 2427x^{8} + 1888x^{6} - 366x^{4} + 700x^{2} + 625 \) x^16 - 12x^14 + 98x^12 - 608x^10 + 2427x^8 + 1888x^6 - 366x^4 + 700*x^2 + 625
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 46 \nu^{14} + 10984 \nu^{12} + 4869 \nu^{10} - 31012 \nu^{8} - 14975 \nu^{6} + 3692 \nu^{4} + \cdots + 887080822 ) / 267466838 \) (46v^14 + 10984v^12 + 4869v^10 - 31012v^8 - 14975v^6 + 3692v^4 - 9025*v^2 + 887080822) / 267466838
\(\beta_{2}\)	\(=\)	\( ( 88901573 \nu^{14} - 1131343676 \nu^{12} + 9539640304 \nu^{10} - 61096184484 \nu^{8} + \cdots + 90100085000 ) / 46806696650 \) (88901573v^14 - 1131343676v^12 + 9539640304v^10 - 61096184484v^8 + 260603918896v^6 - 24479017876v^4 + 8929240657*v^2 + 90100085000) / 46806696650
\(\beta_{3}\)	\(=\)	\( ( - 100243982 \nu^{15} + 1310942259 \nu^{13} - 11054039436 \nu^{11} + 70587360231 \nu^{9} + \cdots - 104403296250 \nu ) / 234033483250 \) (-100243982v^15 + 1310942259v^13 - 11054039436v^11 + 70587360231v^9 - 301974278364v^7 + 28365013809v^5 + 386287043712v^3 - 104403296250v) / 234033483250
\(\beta_{4}\)	\(=\)	\( ( - 8511334 \nu^{14} + 107376168 \nu^{12} - 900906201 \nu^{10} + 5738119348 \nu^{8} + \cdots - 7594410356 ) / 1872267866 \) (-8511334v^14 + 107376168v^12 - 900906201v^10 + 5738119348v^8 - 24243341363v^6 - 683127068v^4 + 1669886725*v^2 - 7594410356) / 1872267866
\(\beta_{5}\)	\(=\)	\( ( 7370156 \nu^{14} - 94041222 \nu^{12} + 792968088 \nu^{10} - 5073733873 \nu^{8} + \cdots + 6257677325 ) / 1231755175 \) (7370156v^14 - 94041222v^12 + 792968088v^10 - 5073733873v^8 + 21662304312v^6 - 2034781122v^4 - 3146885796*v^2 + 6257677325) / 1231755175
\(\beta_{6}\)	\(=\)	\( ( 562150656 \nu^{14} - 7147069697 \nu^{12} + 60265041988 \nu^{10} - 385625932323 \nu^{8} + \cdots + 569191838750 ) / 46806696650 \) (562150656v^14 - 7147069697v^12 + 60265041988v^10 - 385625932323v^8 + 1646320575412v^6 - 154641998347v^4 + 56490278304*v^2 + 569191838750) / 46806696650
\(\beta_{7}\)	\(=\)	\( ( - 634901829 \nu^{15} + 8278918773 \nu^{13} - 69808943892 \nu^{11} + 446544853007 \nu^{9} + \cdots - 659332173750 \nu ) / 468066966500 \) (-634901829v^15 + 8278918773v^13 - 69808943892v^11 + 446544853007v^9 - 1907040912708v^7 + 179131951623v^5 + 2440229361389v^3 - 659332173750v) / 468066966500
\(\beta_{8}\)	\(=\)	\( ( - 781044744 \nu^{14} + 9798111678 \nu^{12} - 81909271112 \nu^{10} + 519921366477 \nu^{8} + \cdots - 630227236425 ) / 46806696650 \) (-781044744v^14 + 9798111678v^12 - 81909271112v^10 + 519921366477v^8 - 2182506988188v^6 - 262448029147v^4 + 320019375804*v^2 - 630227236425) / 46806696650
\(\beta_{9}\)	\(=\)	\( ( - 929099357 \nu^{14} + 11852042984 \nu^{12} - 99938002336 \nu^{10} + 639501846006 \nu^{8} + \cdots - 943895390000 ) / 46806696650 \) (-929099357v^14 + 11852042984v^12 - 99938002336v^10 + 639501846006v^8 - 2730106610464v^6 + 256444065784v^4 + 396622436737*v^2 - 943895390000) / 46806696650
\(\beta_{10}\)	\(=\)	\( ( 1144428498 \nu^{15} - 14797058726 \nu^{13} + 125576013804 \nu^{11} - 808425801909 \nu^{9} + \cdots + 1009835789225 \nu ) / 234033483250 \) (1144428498v^15 - 14797058726v^13 + 125576013804v^11 - 808425801909v^9 + 3494792883146v^7 - 869736666151v^5 - 504251713768v^3 + 1009835789225v) / 234033483250
\(\beta_{11}\)	\(=\)	\( ( - 2652364739 \nu^{15} + 34288646618 \nu^{13} - 290979178297 \nu^{11} + \cdots - 2339349545425 \nu ) / 468066966500 \) (-2652364739v^15 + 34288646618v^13 - 290979178297v^11 + 1873051965937v^9 - 8095938906053v^7 + 2014815612393v^5 + 1168228883974v^3 - 2339349545425v) / 468066966500
\(\beta_{12}\)	\(=\)	\( ( - 62239472 \nu^{15} + 785106535 \nu^{13} - 6587912808 \nu^{11} + 41960228384 \nu^{9} + \cdots - 69269810581 \nu ) / 9361339330 \) (-62239472v^15 + 785106535v^13 - 6587912808v^11 + 41960228384v^9 - 177396339082v^7 - 4995394144v^5 + 12211113800v^3 - 69269810581v) / 9361339330
\(\beta_{13}\)	\(=\)	\( ( - 62239472 \nu^{15} + 785106535 \nu^{13} - 6587912808 \nu^{11} + 41960228384 \nu^{9} + \cdots - 50547131921 \nu ) / 9361339330 \) (-62239472v^15 + 785106535v^13 - 6587912808v^11 + 41960228384v^9 - 177396339082v^7 - 4995394144v^5 + 12211113800v^3 - 50547131921v) / 9361339330
\(\beta_{14}\)	\(=\)	\( ( 144161746 \nu^{15} - 1818438765 \nu^{13} + 15259207419 \nu^{11} - 97190088412 \nu^{9} + \cdots + 160476040633 \nu ) / 18722678660 \) (144161746v^15 - 1818438765v^13 + 15259207419v^11 - 97190088412v^9 + 410972310431v^7 + 11570547092v^5 - 28283907775v^3 + 160476040633v) / 18722678660
\(\beta_{15}\)	\(=\)	\( ( 393116414 \nu^{15} - 4959633785 \nu^{13} + 41610517821 \nu^{11} - 265028831108 \nu^{9} + \cdots + 319291589437 \nu ) / 18722678660 \) (393116414v^15 - 4959633785v^13 + 41610517821v^11 - 265028831108v^9 + 1120558715009v^7 + 31551865228v^5 - 77127731225v^3 + 319291589437v) / 18722678660

\(\nu\)	\(=\)	\( ( \beta_{13} - \beta_{12} ) / 2 \) (b13 - b12) / 2
\(\nu^{2}\)	\(=\)	\( \beta_{9} + 3\beta_{5} + \beta_{2} + 3 \) b9 + 3*b5 + b2 + 3
\(\nu^{3}\)	\(=\)	\( ( -6\beta_{14} - 7\beta_{12} + 6\beta_{11} + 7\beta_{10} + 2\beta_{7} - 5\beta_{3} ) / 2 \) (-6b14 - 7b12 + 6b11 + 7b10 + 2b7 - 5b3) / 2
\(\nu^{4}\)	\(=\)	\( 4\beta_{9} - 2\beta_{8} + 13\beta_{5} + 12\beta_{4} + 12\beta_{2} + 4\beta_1 \) 4b9 - 2b8 + 13b5 + 12b4 + 12b2 + 4b1
\(\nu^{5}\)	\(=\)	\( ( -4\beta_{15} - 13\beta_{13} + 60\beta_{11} + 69\beta_{10} + 4\beta_{7} - 13\beta_{3} ) / 2 \) (-4b15 - 13b13 + 60b11 + 69b10 + 4b7 - 13b3) / 2
\(\nu^{6}\)	\(=\)	\( -16\beta_{8} + 16\beta_{6} + 101\beta_{4} - \beta _1 + 3 \) -16b8 + 16b6 + 101*b4 - b1 + 3
\(\nu^{7}\)	\(=\)	\( ( 34\beta_{15} + 434\beta_{14} + 107\beta_{13} + 503\beta_{12} ) / 2 \) (34b15 + 434b14 + 107b13 + 503b12) / 2
\(\nu^{8}\)	\(=\)	\( 224\beta_{9} + 100\beta_{6} + 743\beta_{5} - 632\beta_{2} + 743 \) 224b9 + 100b6 + 743b5 - 632b2 + 743
\(\nu^{9}\)	\(=\)	\( ( 2280\beta_{14} + 2641\beta_{12} - 2280\beta_{11} - 2641\beta_{10} + 648\beta_{7} - 2047\beta_{3} ) / 2 \) (2280b14 + 2641b12 - 2280b11 - 2641b10 + 648b7 - 2047b3) / 2
\(\nu^{10}\)	\(=\)	\( 2779\beta_{9} + 408\beta_{8} + 9217\beta_{5} - 2577\beta_{4} - 2577\beta_{2} + 2779\beta_1 \) 2779b9 + 408b8 + 9217b5 - 2577b4 - 2577b2 + 2779b1
\(\nu^{11}\)	\(=\)	\( ( -6374\beta_{15} - 20131\beta_{13} - 5566\beta_{11} - 6447\beta_{10} + 6374\beta_{7} - 20131\beta_{3} ) / 2 \) (-6374b15 - 20131b13 - 5566b11 - 6447b10 + 6374b7 - 20131b3) / 2
\(\nu^{12}\)	\(=\)	\( -202\beta_{8} + 202\beta_{6} + 1276\beta_{4} + 23116\beta _1 - 76667 \) -202b8 + 202b6 + 1276b4 + 23116b1 - 76667
\(\nu^{13}\)	\(=\)	\( ( -45828\beta_{15} + 51740\beta_{14} - 144739\beta_{13} + 59931\beta_{12} ) / 2 \) (-45828b15 + 51740b14 - 144739b13 + 59931b12) / 2
\(\nu^{14}\)	\(=\)	\( -143261\beta_{9} + 24392\beta_{6} - 475143\beta_{5} - 154075\beta_{2} - 475143 \) -143261b9 + 24392b6 - 475143b5 - 154075b2 - 475143
\(\nu^{15}\)	\(=\)	\( ( 951606 \beta_{14} + 1102257 \beta_{12} - 951606 \beta_{11} - 1102257 \beta_{10} + \cdots + 750851 \beta_{3} ) / 2 \) (951606b14 + 1102257b12 - 951606b11 - 1102257b10 - 237738b7 + 750851b3) / 2

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(1\)	\(-1 - \beta_{5}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 65.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 8 T^{6} + 59 T^{4} + \cdots + 25)^{2} \) (T^8 - 8T^6 + 59T^4 - 40*T^2 + 25)^2
$5$	\( (T^{8} - 14 T^{6} + \cdots + 25)^{2} \) (T^8 - 14T^6 + 191T^4 - 70*T^2 + 25)^2
$7$	\( (T^{4} - 8 T^{2} + 49)^{4} \) (T^4 - 8*T^2 + 49)^4
$11$	\( (T^{8} - 32 T^{6} + \cdots + 60025)^{2} \) (T^8 - 32T^6 + 779T^4 - 7840*T^2 + 60025)^2
$13$	\( (T^{4} + 16 T^{2} + 20)^{4} \) (T^4 + 16*T^2 + 20)^4
$17$	\( (T^{4} - 4 T^{3} + 23 T^{2} + \cdots + 49)^{4} \) (T^4 - 4T^3 + 23T^2 + 28*T + 49)^4
$19$	\( (T^{8} - 40 T^{6} + \cdots + 15625)^{2} \) (T^8 - 40T^6 + 1475T^4 - 5000*T^2 + 15625)^2
$23$	\( (T^{8} + 36 T^{6} + \cdots + 2401)^{2} \) (T^8 + 36T^6 + 1247T^4 + 1764*T^2 + 2401)^2
$29$	\( (T^{4} + 64 T^{2} + 980)^{4} \) (T^4 + 64*T^2 + 980)^4
$31$	\( (T^{8} + 100 T^{6} + \cdots + 5764801)^{2} \) (T^8 + 100T^6 + 7599T^4 + 240100*T^2 + 5764801)^2
$37$	\( (T^{8} - 170 T^{6} + \cdots + 37515625)^{2} \) (T^8 - 170T^6 + 22775T^4 - 1041250*T^2 + 37515625)^2
$41$	\( (T^{2} - 10 T + 14)^{8} \) (T^2 - 10*T + 14)^8
$43$	\( (T^{4} + 128 T^{2} + 3920)^{4} \) (T^4 + 128*T^2 + 3920)^4
$47$	\( (T^{8} + 92 T^{6} + \cdots + 1500625)^{2} \) (T^8 + 92T^6 + 7239T^4 + 112700*T^2 + 1500625)^2
$53$	\( (T^{8} - 170 T^{6} + \cdots + 37515625)^{2} \) (T^8 - 170T^6 + 22775T^4 - 1041250*T^2 + 37515625)^2
$59$	\( (T^{8} - 160 T^{6} + \cdots + 37515625)^{2} \) (T^8 - 160T^6 + 19475T^4 - 980000*T^2 + 37515625)^2
$61$	\( (T^{8} - 26 T^{6} + \cdots + 15625)^{2} \) (T^8 - 26T^6 + 551T^4 - 3250*T^2 + 15625)^2
$67$	\( (T^{8} - 352 T^{6} + \cdots + 878826025)^{2} \) (T^8 - 352T^6 + 94259T^4 - 10435040*T^2 + 878826025)^2
$71$	\( (T^{4} - 48 T^{2} + 400)^{4} \) (T^4 - 48*T^2 + 400)^4
$73$	\( (T^{4} + 6 T^{3} + \cdots + 1225)^{4} \) (T^4 + 6T^3 + 71T^2 - 210*T + 1225)^4
$79$	\( (T^{8} + 12 T^{6} + \cdots + 625)^{2} \) (T^8 + 12T^6 + 119T^4 + 300*T^2 + 625)^2
$83$	\( (T^{4} + 128 T^{2} + 1280)^{4} \) (T^4 + 128*T^2 + 1280)^4
$89$	\( (T^{2} + 7 T + 49)^{8} \) (T^2 + 7*T + 49)^8
$97$	\( (T^{2} + 2 T - 98)^{8} \) (T^2 + 2*T - 98)^8
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