LMFDB - Newform orbit 896.2.t.b

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} - 16x^{14} + 188x^{12} - 912x^{10} + 3212x^{8} - 5856x^{6} + 7472x^{4} - 352x^{2} + 16 $ x^16 - 16x^14 + 188x^12 - 912x^10 + 3212x^8 - 5856x^6 + 7472x^4 - 352*x^2 + 16
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_{4} - \beta_1) q^{3} + \beta_{7} q^{5} + (\beta_{15} + \beta_{11}) q^{7} + ( - \beta_{9} + \beta_{5} + 1) q^{9}+O(q^{10}) $ q + (-b4 - b1) * q^3 + b7 * q^5 + (b15 + b11) * q^7 + (-b9 + b5 + 1) * q^9	$ q + ( - \beta_{4} - \beta_1) q^{3} + \beta_{7} q^{5} + (\beta_{15} + \beta_{11}) q^{7} + ( - \beta_{9} + \beta_{5} + 1) q^{9} + (2 \beta_{4} + \beta_1) q^{11} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_{3}) q^{13}+ \cdots + ( - 7 \beta_{13} + 4 \beta_{12}) q^{99}+O(q^{100}) $ q + (-b4 - b1) * q^3 + b7 * q^5 + (b15 + b11) * q^7 + (-b9 + b5 + 1) * q^9 + (2b4 + b1) q^11 + (-b8 - b7 + 2b6 + b3) q^13 + (b15 + b14 + 2b11 - b10) q^15 + (-b9 + 2b5 - b2) q^17 + (-b12 + b1) * q^19 + (-b8 + b7 + b6 + b3) * q^21 + (b15 + b10) * q^23 + (2b9 - 2b5 + 2b2) q^25 + (2b13 + b12) q^27 + (b8 + b7 + 2b6 + b3) q^29 + (-b11 - b10) * q^31 + (2b9 - 5b5 - 5) * q^33 + (-5b13 + 3b12 - 4b4 - b1) q^35 + (2b7 + b6 - b3) q^37 + (-3b15 - 3b14) * q^39 + (-3b2 - 3) q^41 - 2b12 q^43 + (-3b8 + b6 + 2b3) * q^45 + (-b15 + 3b14 - 3b11 + 2b10) q^47 + (2b9 + 3b2) * q^49 + (6b13 - 3b12 + 6b4 + 3b1) * q^51 + (-2b8 + b6 + 2b3) * q^53 + (-2b15 - b14 - 3b11 + 2b10) q^55 - 3 * q^57 + (b4 - b1) * q^59 + (2b7 + b6 - b3) q^61 + (-b14 + b11 - 3b10) q^63 + (-5b9 + 7b5 + 7) * q^65 + (3b4 - 3b1) * q^67 + (3b8 + 3b7) * q^69 + (-2b14 - 2b11) * q^71 + (-2b9 - b5 - 2b2) * q^73 + (-10b13 + 4b12 - 10b4 - 4b1) * q^75 + (2b8 - b7 - b3) q^77 + (-3b15 + 2b14 - 2b11 - b10) q^79 + (b9 - 4b5 + b2) q^81 + 8b13 q^83 + (-4b8 - 4b7 + 2b6 + b3) q^85 + (b15 + b14 + 2b11 + 2b10) * q^87 + (9b5 + 9) q^89 + (3b13 - 6b12 - 6b4 + 2b1) * q^91 + (-b6 + b3) * q^93 + (-b15 - b14 - b11 - b10) * q^95 + (-3b2 - 1) q^97 + (-7b13 + 4b12) * 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} - 40 q^{33} - 48 q^{41} - 48 q^{57} + 56 q^{65} + 8 q^{73} + 32 q^{81} + 72 q^{89} - 16 q^{97}+O(q^{100}) $ 16 * q + 8 * q^9 - 16 * q^17 + 16 * q^25 - 40 * q^33 - 48 * q^41 - 48 * q^57 + 56 * q^65 + 8 * q^73 + 32 * q^81 + 72 * q^89 - 16 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 16x^{14} + 188x^{12} - 912x^{10} + 3212x^{8} - 5856x^{6} + 7472x^{4} - 352x^{2} + 16 $ x^16 - 16*x^14 + 188*x^12 - 912*x^10 + 3212*x^8 - 5856*x^6 + 7472*x^4 - 352*x^2 + 16 :

$\beta_{1}$	$=$	$ ( 102068 \nu^{15} - 1489401 \nu^{13} + 17081380 \nu^{11} - 68349536 \nu^{9} + \cdots + 1005475528 \nu ) / 187690320 $ (102068v^15 - 1489401v^13 + 17081380v^11 - 68349536v^9 + 220582232v^7 - 178919200v^5 + 8429616v^3 + 1005475528v) / 187690320
$\beta_{2}$	$=$	$ ( - 340 \nu^{14} + 5153 \nu^{12} - 56900 \nu^{10} + 227680 \nu^{8} - 529756 \nu^{6} + 596000 \nu^{4} + \cdots + 952576 ) / 359560 $ (-340v^14 + 5153v^12 - 56900v^10 + 227680v^8 - 529756v^6 + 596000v^4 - 28080*v^2 + 952576) / 359560
$\beta_{3}$	$=$	$ ( 108086 \nu^{14} - 1626453 \nu^{12} + 18088510 \nu^{10} - 72379472 \nu^{8} + 181644836 \nu^{6} + \cdots + 122262304 ) / 93845160 $ (108086v^14 - 1626453v^12 + 18088510v^10 - 72379472v^8 + 181644836v^6 - 189468400v^4 + 8926632*v^2 + 122262304) / 93845160
$\beta_{4}$	$=$	$ ( 105077 \nu^{15} - 1557927 \nu^{13} + 17584945 \nu^{11} - 70364504 \nu^{9} + 201113534 \nu^{7} + \cdots + 610791496 \nu ) / 93845160 $ (105077v^15 - 1557927v^13 + 17584945v^11 - 70364504v^9 + 201113534v^7 - 184193800v^5 + 8678124v^3 + 610791496v) / 93845160
$\beta_{5}$	$=$	$ ( 140500 \nu^{14} - 2238327 \nu^{12} + 26273240 \nu^{10} - 126517195 \nu^{8} + 444808504 \nu^{6} + \cdots - 48657124 ) / 46922580 $ (140500v^14 - 2238327v^12 + 26273240v^10 - 126517195v^8 + 444808504v^6 - 801078740v^4 + 1032859800*v^2 - 48657124) / 46922580
$\beta_{6}$	$=$	$ ( 296219 \nu^{14} - 5019732 \nu^{12} + 59923690 \nu^{10} - 317048708 \nu^{8} + 1139109284 \nu^{6} + \cdots - 127412624 ) / 93845160 $ (296219v^14 - 5019732v^12 + 59923690v^10 - 317048708v^8 + 1139109284v^6 - 2189146300v^4 + 2704571568*v^2 - 127412624) / 93845160
$\beta_{7}$	$=$	$ ( - 471227 \nu^{14} + 7720356 \nu^{12} - 91511920 \nu^{10} + 463223894 \nu^{8} - 1670484392 \nu^{6} + \cdots + 386888912 ) / 93845160 $ (-471227v^14 + 7720356v^12 - 91511920v^10 + 463223894v^8 - 1670484392v^6 + 3216324640v^4 - 4180065504*v^2 + 386888912) / 93845160
$\beta_{8}$	$=$	$ ( - 132407 \nu^{14} + 2110938 \nu^{12} - 24688840 \nu^{10} + 118199582 \nu^{8} - 405866588 \nu^{6} + \cdots - 36454792 ) / 18769032 $ (-132407v^14 + 2110938v^12 - 24688840v^10 + 118199582v^8 - 405866588v^6 + 710159968v^4 - 839164800*v^2 - 36454792) / 18769032
$\beta_{9}$	$=$	$ ( - 719695 \nu^{14} + 11260431 \nu^{12} - 132173720 \nu^{10} + 622709380 \nu^{8} - 2237713912 \nu^{6} + \cdots + 8726032 ) / 93845160 $ (-719695v^14 + 11260431v^12 - 132173720v^10 + 622709380v^8 - 2237713912v^6 + 4030015220v^4 - 5455535040*v^2 + 8726032) / 93845160
$\beta_{10}$	$=$	$ ( - 1158248 \nu^{15} + 18554799 \nu^{13} - 218217550 \nu^{11} + 1064053226 \nu^{9} + \cdots + 2489823248 \nu ) / 187690320 $ (-1158248v^15 + 18554799v^13 - 218217550v^11 + 1064053226v^9 - 3782503868v^7 + 7189064260v^5 - 9632902056v^3 + 2489823248v) / 187690320
$\beta_{11}$	$=$	$ ( - 250524 \nu^{15} + 3931007 \nu^{13} - 45989485 \nu^{11} + 215835303 \nu^{9} + \cdots - 336522316 \nu ) / 31281720 $ (-250524v^15 + 3931007v^13 - 45989485v^11 + 215835303v^9 - 755167934v^7 + 1298941110v^5 - 1610231068v^3 - 336522316v) / 31281720
$\beta_{12}$	$=$	$ ( 341888 \nu^{15} - 5449275 \nu^{13} + 63963000 \nu^{11} - 308221986 \nu^{9} + 1082899800 \nu^{7} + \cdots - 4222800 \nu ) / 20854480 $ (341888v^15 - 5449275v^13 + 63963000v^11 - 308221986v^9 + 1082899800v^7 - 1950250500v^5 + 2462076416v^3 - 4222800v) / 20854480
$\beta_{13}$	$=$	$ ( - 4697 \nu^{15} + 75114 \nu^{13} - 881680 \nu^{11} + 4265244 \nu^{9} - 14926928 \nu^{7} + \cdots + 58208 \nu ) / 234320 $ (-4697v^15 + 75114v^13 - 881680v^11 + 4265244v^9 - 14926928v^7 + 26882680v^5 - 33630344v^3 + 58208v) / 234320
$\beta_{14}$	$=$	$ ( 2445845 \nu^{15} - 39032639 \nu^{13} + 458465390 \nu^{11} - 2215644680 \nu^{9} + \cdots - 709384648 \nu ) / 62563440 $ (2445845v^15 - 39032639v^13 + 458465390v^11 - 2215644680v^9 + 7808740588v^7 - 14185326640v^5 + 18159238720v^3 - 709384648v) / 62563440
$\beta_{15}$	$=$	$ ( 7547689 \nu^{15} - 120213771 \nu^{13} + 1410566060 \nu^{11} - 6787663048 \nu^{9} + \cdots - 531190312 \nu ) / 187690320 $ (7547689v^15 - 120213771v^13 + 1410566060v^11 - 6787663048v^9 + 23828448832v^7 - 42924367520v^5 + 54495072408v^3 - 531190312v) / 187690320

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{14} - \beta_{4} ) / 2 $ (b15 - b14 - b4) / 2
$\nu^{2}$	$=$	$ \beta_{9} - \beta_{6} + 4\beta_{5} - \beta_{3} + 4 $ b9 - b6 + 4*b5 - b3 + 4
$\nu^{3}$	$=$	$ 3\beta_{15} - \beta_{14} + 4\beta_{13} + 3\beta_{12} + 4\beta_{11} + 3\beta_{10} $ 3b15 - b14 + 4b13 + 3b12 + 4b11 + 3*b10
$\nu^{4}$	$=$	$ 10\beta_{9} - 2\beta_{8} - 4\beta_{7} - 12\beta_{6} + 30\beta_{5} + 10\beta_{2} $ 10b9 - 2b8 - 4b7 - 12b6 + 30b5 + 10b2
$\nu^{5}$	$=$	$ -12\beta_{15} + 25\beta_{14} + 37\beta_{13} + 40\beta_{12} + 25\beta_{11} + 37\beta_{10} + 37\beta_{4} - 40\beta_1 $ -12b15 + 25b14 + 37b13 + 40b12 + 25b11 + 37b10 + 37b4 - 40b1
$\nu^{6}$	$=$	$ 26\beta_{8} - 26\beta_{7} + 128\beta_{3} + 100\beta_{2} - 274 $ 26b8 - 26b7 + 128b3 + 100b2 - 274
$\nu^{7}$	$=$	$ -364\beta_{15} + 364\beta_{14} - 126\beta_{11} + 126\beta_{10} + 368\beta_{4} - 434\beta_1 $ -364b15 + 364b14 - 126b11 + 126b10 + 368b4 - 434b1
$\nu^{8}$	$=$	$ -1008\beta_{9} + 560\beta_{8} + 280\beta_{7} + 1320\beta_{6} - 2692\beta_{5} + 1320\beta_{3} - 2692 $ -1008b9 + 560b8 + 280b7 + 1320b6 - 2692b5 + 1320b3 - 2692
$\nu^{9}$	$=$	$ -2370\beta_{15} + 1288\beta_{14} - 3722\beta_{13} - 4488\beta_{12} - 3658\beta_{11} - 2370\beta_{10} $ -2370b15 + 1288b14 - 3722b13 - 4488b12 - 3658b11 - 2370b10
$\nu^{10}$	$=$	$ -10204\beta_{9} + 2888\beta_{8} + 5776\beta_{7} + 13468\beta_{6} - 27064\beta_{5} - 10204\beta_{2} $ -10204b9 + 2888b8 + 5776b7 + 13468b6 - 27064b5 - 10204b2
$\nu^{11}$	$=$	$ 13092 \beta_{15} - 23924 \beta_{14} - 37768 \beta_{13} - 45804 \beta_{12} - 23924 \beta_{11} + \cdots + 45804 \beta_1 $ 13092b15 - 23924b14 - 37768b13 - 45804b12 - 23924b11 - 37016b10 - 37768b4 + 45804b1
$\nu^{12}$	$=$	$ -29448\beta_{8} + 29448\beta_{7} - 136944\beta_{3} - 103480\beta_{2} + 273960 $ -29448b8 + 29448b7 - 136944b3 - 103480b2 + 273960
$\nu^{13}$	$=$	$ 375396\beta_{15} - 375396\beta_{14} + 132928\beta_{11} - 132928\beta_{10} - 383428\beta_{4} + 465712\beta_1 $ 375396b15 - 375396b14 + 132928b11 - 132928b10 - 383428b4 + 465712b1
$\nu^{14}$	$=$	$ 1050112 \beta_{9} - 598640 \beta_{8} - 299320 \beta_{7} - 1390928 \beta_{6} + 2778808 \beta_{5} + \cdots + 2778808 $ 1050112b9 - 598640b8 - 299320b7 - 1390928b6 + 2778808b5 - 1390928b3 + 2778808
$\nu^{15}$	$=$	$ 2460264 \beta_{15} - 1349432 \beta_{14} + 3892688 \beta_{13} + 4729928 \beta_{12} + \cdots + 2460264 \beta_{10} $ 2460264b15 - 1349432b14 + 3892688b13 + 4729928b12 + 3809696b11 + 2460264b10

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

−1.41280	+	0.815681i
0.188057	−	0.108574i
−1.53456	+	0.885978i
2.75930	−	1.59308i
1.53456	−	0.885978i
−2.75930	+	1.59308i
1.41280	−	0.815681i
−0.188057	+	0.108574i
−1.41280	−	0.815681i
0.188057	+	0.108574i
−1.53456	−	0.885978i
2.75930	+	1.59308i
1.53456	+	0.885978i
−2.75930	−	1.59308i
1.41280	+	0.815681i
−0.188057	−	0.108574i

−2.23256

1.28897i

−3.03622

−

1.75296i

2.47906

0.924256i

1.82288

−

3.15731i

65.2

−2.23256

1.28897i

3.03622

1.75296i

−2.47906

−

0.924256i

1.82288

−

3.15731i

65.3

−1.00781

0.581861i

−1.13198

−

0.653548i

−0.924256

2.47906i

−0.822876

1.42526i

65.4

−1.00781

0.581861i

1.13198

0.653548i

0.924256

−

2.47906i

−0.822876

1.42526i

65.5

1.00781

−

0.581861i

−1.13198

−

0.653548i

0.924256

−

2.47906i

−0.822876

1.42526i

65.6

1.00781

−

0.581861i

1.13198

0.653548i

−0.924256

2.47906i

−0.822876

1.42526i

65.7

2.23256

−

1.28897i

−3.03622

−

1.75296i

−2.47906

−

0.924256i

1.82288

−

3.15731i

65.8

2.23256

−

1.28897i

3.03622

1.75296i

2.47906

0.924256i

1.82288

−

3.15731i

193.1

−2.23256

−

1.28897i

−3.03622

1.75296i

2.47906

−

0.924256i

1.82288

3.15731i

193.2

−2.23256

−

1.28897i

3.03622

−

1.75296i

−2.47906

0.924256i

1.82288

3.15731i

193.3

−1.00781

−

0.581861i

−1.13198

0.653548i

−0.924256

−

2.47906i

−0.822876

−

1.42526i

193.4

−1.00781

−

0.581861i

1.13198

−

0.653548i

0.924256

2.47906i

−0.822876

−

1.42526i

193.5

1.00781

0.581861i

−1.13198

0.653548i

0.924256

2.47906i

−0.822876

−

1.42526i

193.6

1.00781

0.581861i

1.13198

−

0.653548i

−0.924256

−

2.47906i

−0.822876

−

1.42526i

193.7

2.23256

1.28897i

−3.03622

1.75296i

−2.47906

0.924256i

1.82288

3.15731i

193.8

2.23256

1.28897i

3.03622

−

1.75296i

2.47906

−

0.924256i

1.82288

3.15731i

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.b	✓	16
4.b	odd	2	1	inner	896.2.t.b	✓	16
7.c	even	3	1	inner	896.2.t.b	✓	16
8.b	even	2	1	inner	896.2.t.b	✓	16
8.d	odd	2	1	inner	896.2.t.b	✓	16
28.g	odd	6	1	inner	896.2.t.b	✓	16
56.k	odd	6	1	inner	896.2.t.b	✓	16
56.p	even	6	1	inner	896.2.t.b	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
896.2.t.b	✓	16	1.a	even	1	1	trivial
896.2.t.b	✓	16	4.b	odd	2	1	inner
896.2.t.b	✓	16	7.c	even	3	1	inner
896.2.t.b	✓	16	8.b	even	2	1	inner
896.2.t.b	✓	16	8.d	odd	2	1	inner
896.2.t.b	✓	16	28.g	odd	6	1	inner
896.2.t.b	✓	16	56.k	odd	6	1	inner
896.2.t.b	✓	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} + 55T_{3}^{4} - 72T_{3}^{2} + 81 $ T3^8 - 8*T3^6 + 55*T3^4 - 72*T3^2 + 81 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} + 55 T^{4} + \cdots + 81)^{2} $ (T^8 - 8T^6 + 55T^4 - 72*T^2 + 81)^2
$5$	$ (T^{8} - 14 T^{6} + \cdots + 441)^{2} $ (T^8 - 14T^6 + 175T^4 - 294*T^2 + 441)^2
$7$	$ (T^{8} - 14 T^{4} + 2401)^{2} $ (T^8 - 14*T^4 + 2401)^2
$11$	$ (T^{8} - 16 T^{6} + 255 T^{4} + \cdots + 1)^{2} $ (T^8 - 16T^6 + 255T^4 - 16*T^2 + 1)^2
$13$	$ (T^{4} + 56 T^{2} + 756)^{4} $ (T^4 + 56*T^2 + 756)^4
$17$	$ (T^{4} + 4 T^{3} + 19 T^{2} + \cdots + 9)^{4} $ (T^4 + 4T^3 + 19T^2 - 12*T + 9)^4
$19$	$ (T^{8} - 8 T^{6} + 55 T^{4} + \cdots + 81)^{2} $ (T^8 - 8T^6 + 55T^4 - 72*T^2 + 81)^2
$23$	$ (T^{8} + 28 T^{6} + \cdots + 35721)^{2} $ (T^8 + 28T^6 + 595T^4 + 5292*T^2 + 35721)^2
$29$	$ (T^{4} + 56 T^{2} + 84)^{4} $ (T^4 + 56*T^2 + 84)^4
$31$	$ (T^{8} + 28 T^{6} + \cdots + 441)^{2} $ (T^8 + 28T^6 + 763T^4 + 588*T^2 + 441)^2
$37$	$ (T^{8} - 98 T^{6} + \cdots + 2893401)^{2} $ (T^8 - 98T^6 + 7903T^4 - 166698*T^2 + 2893401)^2
$41$	$ (T^{2} + 6 T - 54)^{8} $ (T^2 + 6*T - 54)^8
$43$	$ (T^{4} + 32 T^{2} + 144)^{4} $ (T^4 + 32*T^2 + 144)^4
$47$	$ (T^{8} + 196 T^{6} + \cdots + 85766121)^{2} $ (T^8 + 196T^6 + 29155T^4 + 1815156*T^2 + 85766121)^2
$53$	$ (T^{8} - 98 T^{6} + \cdots + 1058841)^{2} $ (T^8 - 98T^6 + 8575T^4 - 100842*T^2 + 1058841)^2
$59$	$ (T^{8} - 16 T^{6} + 255 T^{4} + \cdots + 1)^{2} $ (T^8 - 16T^6 + 255T^4 - 16*T^2 + 1)^2
$61$	$ (T^{8} - 98 T^{6} + \cdots + 2893401)^{2} $ (T^8 - 98T^6 + 7903T^4 - 166698*T^2 + 2893401)^2
$67$	$ (T^{8} - 144 T^{6} + \cdots + 6561)^{2} $ (T^8 - 144T^6 + 20655T^4 - 11664*T^2 + 6561)^2
$71$	$ (T^{4} - 112 T^{2} + 3024)^{4} $ (T^4 - 112*T^2 + 3024)^4
$73$	$ (T^{4} - 2 T^{3} + \cdots + 729)^{4} $ (T^4 - 2T^3 + 31T^2 + 54*T + 729)^4
$79$	$ (T^{8} + 196 T^{6} + \cdots + 1058841)^{2} $ (T^8 + 196T^6 + 37387T^4 + 201684*T^2 + 1058841)^2
$83$	$ (T^{2} + 128)^{8} $ (T^2 + 128)^8
$89$	$ (T^{2} - 9 T + 81)^{8} $ (T^2 - 9*T + 81)^8
$97$	$ (T^{2} + 2 T - 62)^{8} $ (T^2 + 2*T - 62)^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)

\(f(q)\)	\(=\)	\( q + ( - \beta_{4} - \beta_1) q^{3} + \beta_{7} q^{5} + (\beta_{15} + \beta_{11}) q^{7} + ( - \beta_{9} + \beta_{5} + 1) q^{9}+O(q^{10}) \) q + (-b4 - b1) * q^3 + b7 * q^5 + (b15 + b11) * q^7 + (-b9 + b5 + 1) * q^9	\( q + ( - \beta_{4} - \beta_1) q^{3} + \beta_{7} q^{5} + (\beta_{15} + \beta_{11}) q^{7} + ( - \beta_{9} + \beta_{5} + 1) q^{9} + (2 \beta_{4} + \beta_1) q^{11} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_{3}) q^{13}+ \cdots + ( - 7 \beta_{13} + 4 \beta_{12}) q^{99}+O(q^{100}) \) q + (-b4 - b1) * q^3 + b7 * q^5 + (b15 + b11) * q^7 + (-b9 + b5 + 1) * q^9 + (2b4 + b1) q^11 + (-b8 - b7 + 2b6 + b3) q^13 + (b15 + b14 + 2b11 - b10) q^15 + (-b9 + 2b5 - b2) q^17 + (-b12 + b1) * q^19 + (-b8 + b7 + b6 + b3) * q^21 + (b15 + b10) * q^23 + (2b9 - 2b5 + 2b2) q^25 + (2b13 + b12) q^27 + (b8 + b7 + 2b6 + b3) q^29 + (-b11 - b10) * q^31 + (2b9 - 5b5 - 5) * q^33 + (-5b13 + 3b12 - 4b4 - b1) q^35 + (2b7 + b6 - b3) q^37 + (-3b15 - 3b14) * q^39 + (-3b2 - 3) q^41 - 2b12 q^43 + (-3b8 + b6 + 2b3) * q^45 + (-b15 + 3b14 - 3b11 + 2b10) q^47 + (2b9 + 3b2) * q^49 + (6b13 - 3b12 + 6b4 + 3b1) * q^51 + (-2b8 + b6 + 2b3) * q^53 + (-2b15 - b14 - 3b11 + 2b10) q^55 - 3 * q^57 + (b4 - b1) * q^59 + (2b7 + b6 - b3) q^61 + (-b14 + b11 - 3b10) q^63 + (-5b9 + 7b5 + 7) * q^65 + (3b4 - 3b1) * q^67 + (3b8 + 3b7) * q^69 + (-2b14 - 2b11) * q^71 + (-2b9 - b5 - 2b2) * q^73 + (-10b13 + 4b12 - 10b4 - 4b1) * q^75 + (2b8 - b7 - b3) q^77 + (-3b15 + 2b14 - 2b11 - b10) q^79 + (b9 - 4b5 + b2) q^81 + 8b13 q^83 + (-4b8 - 4b7 + 2b6 + b3) q^85 + (b15 + b14 + 2b11 + 2b10) * q^87 + (9b5 + 9) q^89 + (3b13 - 6b12 - 6b4 + 2b1) * q^91 + (-b6 + b3) * q^93 + (-b15 - b14 - b11 - b10) * q^95 + (-3b2 - 1) q^97 + (-7b13 + 4b12) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9}+O(q^{10}) \) 16 * q + 8 * q^9	\( 16 q + 8 q^{9} - 16 q^{17} + 16 q^{25} - 40 q^{33} - 48 q^{41} - 48 q^{57} + 56 q^{65} + 8 q^{73} + 32 q^{81} + 72 q^{89} - 16 q^{97}+O(q^{100}) \) 16 * q + 8 * q^9 - 16 * q^17 + 16 * q^25 - 40 * q^33 - 48 * q^41 - 48 * q^57 + 56 * q^65 + 8 * q^73 + 32 * q^81 + 72 * q^89 - 16 * q^97

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	896.2.t.b

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:	\( x^{16} - 16x^{14} + 188x^{12} - 912x^{10} + 3212x^{8} - 5856x^{6} + 7472x^{4} - 352x^{2} + 16 \) x^16 - 16x^14 + 188x^12 - 912x^10 + 3212x^8 - 5856x^6 + 7472x^4 - 352*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 102068 \nu^{15} - 1489401 \nu^{13} + 17081380 \nu^{11} - 68349536 \nu^{9} + \cdots + 1005475528 \nu ) / 187690320 \) (102068v^15 - 1489401v^13 + 17081380v^11 - 68349536v^9 + 220582232v^7 - 178919200v^5 + 8429616v^3 + 1005475528v) / 187690320
\(\beta_{2}\)	\(=\)	\( ( - 340 \nu^{14} + 5153 \nu^{12} - 56900 \nu^{10} + 227680 \nu^{8} - 529756 \nu^{6} + 596000 \nu^{4} + \cdots + 952576 ) / 359560 \) (-340v^14 + 5153v^12 - 56900v^10 + 227680v^8 - 529756v^6 + 596000v^4 - 28080*v^2 + 952576) / 359560
\(\beta_{3}\)	\(=\)	\( ( 108086 \nu^{14} - 1626453 \nu^{12} + 18088510 \nu^{10} - 72379472 \nu^{8} + 181644836 \nu^{6} + \cdots + 122262304 ) / 93845160 \) (108086v^14 - 1626453v^12 + 18088510v^10 - 72379472v^8 + 181644836v^6 - 189468400v^4 + 8926632*v^2 + 122262304) / 93845160
\(\beta_{4}\)	\(=\)	\( ( 105077 \nu^{15} - 1557927 \nu^{13} + 17584945 \nu^{11} - 70364504 \nu^{9} + 201113534 \nu^{7} + \cdots + 610791496 \nu ) / 93845160 \) (105077v^15 - 1557927v^13 + 17584945v^11 - 70364504v^9 + 201113534v^7 - 184193800v^5 + 8678124v^3 + 610791496v) / 93845160
\(\beta_{5}\)	\(=\)	\( ( 140500 \nu^{14} - 2238327 \nu^{12} + 26273240 \nu^{10} - 126517195 \nu^{8} + 444808504 \nu^{6} + \cdots - 48657124 ) / 46922580 \) (140500v^14 - 2238327v^12 + 26273240v^10 - 126517195v^8 + 444808504v^6 - 801078740v^4 + 1032859800*v^2 - 48657124) / 46922580
\(\beta_{6}\)	\(=\)	\( ( 296219 \nu^{14} - 5019732 \nu^{12} + 59923690 \nu^{10} - 317048708 \nu^{8} + 1139109284 \nu^{6} + \cdots - 127412624 ) / 93845160 \) (296219v^14 - 5019732v^12 + 59923690v^10 - 317048708v^8 + 1139109284v^6 - 2189146300v^4 + 2704571568*v^2 - 127412624) / 93845160
\(\beta_{7}\)	\(=\)	\( ( - 471227 \nu^{14} + 7720356 \nu^{12} - 91511920 \nu^{10} + 463223894 \nu^{8} - 1670484392 \nu^{6} + \cdots + 386888912 ) / 93845160 \) (-471227v^14 + 7720356v^12 - 91511920v^10 + 463223894v^8 - 1670484392v^6 + 3216324640v^4 - 4180065504*v^2 + 386888912) / 93845160
\(\beta_{8}\)	\(=\)	\( ( - 132407 \nu^{14} + 2110938 \nu^{12} - 24688840 \nu^{10} + 118199582 \nu^{8} - 405866588 \nu^{6} + \cdots - 36454792 ) / 18769032 \) (-132407v^14 + 2110938v^12 - 24688840v^10 + 118199582v^8 - 405866588v^6 + 710159968v^4 - 839164800*v^2 - 36454792) / 18769032
\(\beta_{9}\)	\(=\)	\( ( - 719695 \nu^{14} + 11260431 \nu^{12} - 132173720 \nu^{10} + 622709380 \nu^{8} - 2237713912 \nu^{6} + \cdots + 8726032 ) / 93845160 \) (-719695v^14 + 11260431v^12 - 132173720v^10 + 622709380v^8 - 2237713912v^6 + 4030015220v^4 - 5455535040*v^2 + 8726032) / 93845160
\(\beta_{10}\)	\(=\)	\( ( - 1158248 \nu^{15} + 18554799 \nu^{13} - 218217550 \nu^{11} + 1064053226 \nu^{9} + \cdots + 2489823248 \nu ) / 187690320 \) (-1158248v^15 + 18554799v^13 - 218217550v^11 + 1064053226v^9 - 3782503868v^7 + 7189064260v^5 - 9632902056v^3 + 2489823248v) / 187690320
\(\beta_{11}\)	\(=\)	\( ( - 250524 \nu^{15} + 3931007 \nu^{13} - 45989485 \nu^{11} + 215835303 \nu^{9} + \cdots - 336522316 \nu ) / 31281720 \) (-250524v^15 + 3931007v^13 - 45989485v^11 + 215835303v^9 - 755167934v^7 + 1298941110v^5 - 1610231068v^3 - 336522316v) / 31281720
\(\beta_{12}\)	\(=\)	\( ( 341888 \nu^{15} - 5449275 \nu^{13} + 63963000 \nu^{11} - 308221986 \nu^{9} + 1082899800 \nu^{7} + \cdots - 4222800 \nu ) / 20854480 \) (341888v^15 - 5449275v^13 + 63963000v^11 - 308221986v^9 + 1082899800v^7 - 1950250500v^5 + 2462076416v^3 - 4222800v) / 20854480
\(\beta_{13}\)	\(=\)	\( ( - 4697 \nu^{15} + 75114 \nu^{13} - 881680 \nu^{11} + 4265244 \nu^{9} - 14926928 \nu^{7} + \cdots + 58208 \nu ) / 234320 \) (-4697v^15 + 75114v^13 - 881680v^11 + 4265244v^9 - 14926928v^7 + 26882680v^5 - 33630344v^3 + 58208v) / 234320
\(\beta_{14}\)	\(=\)	\( ( 2445845 \nu^{15} - 39032639 \nu^{13} + 458465390 \nu^{11} - 2215644680 \nu^{9} + \cdots - 709384648 \nu ) / 62563440 \) (2445845v^15 - 39032639v^13 + 458465390v^11 - 2215644680v^9 + 7808740588v^7 - 14185326640v^5 + 18159238720v^3 - 709384648v) / 62563440
\(\beta_{15}\)	\(=\)	\( ( 7547689 \nu^{15} - 120213771 \nu^{13} + 1410566060 \nu^{11} - 6787663048 \nu^{9} + \cdots - 531190312 \nu ) / 187690320 \) (7547689v^15 - 120213771v^13 + 1410566060v^11 - 6787663048v^9 + 23828448832v^7 - 42924367520v^5 + 54495072408v^3 - 531190312v) / 187690320

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{14} - \beta_{4} ) / 2 \) (b15 - b14 - b4) / 2
\(\nu^{2}\)	\(=\)	\( \beta_{9} - \beta_{6} + 4\beta_{5} - \beta_{3} + 4 \) b9 - b6 + 4*b5 - b3 + 4
\(\nu^{3}\)	\(=\)	\( 3\beta_{15} - \beta_{14} + 4\beta_{13} + 3\beta_{12} + 4\beta_{11} + 3\beta_{10} \) 3b15 - b14 + 4b13 + 3b12 + 4b11 + 3*b10
\(\nu^{4}\)	\(=\)	\( 10\beta_{9} - 2\beta_{8} - 4\beta_{7} - 12\beta_{6} + 30\beta_{5} + 10\beta_{2} \) 10b9 - 2b8 - 4b7 - 12b6 + 30b5 + 10b2
\(\nu^{5}\)	\(=\)	\( -12\beta_{15} + 25\beta_{14} + 37\beta_{13} + 40\beta_{12} + 25\beta_{11} + 37\beta_{10} + 37\beta_{4} - 40\beta_1 \) -12b15 + 25b14 + 37b13 + 40b12 + 25b11 + 37b10 + 37b4 - 40b1
\(\nu^{6}\)	\(=\)	\( 26\beta_{8} - 26\beta_{7} + 128\beta_{3} + 100\beta_{2} - 274 \) 26b8 - 26b7 + 128b3 + 100b2 - 274
\(\nu^{7}\)	\(=\)	\( -364\beta_{15} + 364\beta_{14} - 126\beta_{11} + 126\beta_{10} + 368\beta_{4} - 434\beta_1 \) -364b15 + 364b14 - 126b11 + 126b10 + 368b4 - 434b1
\(\nu^{8}\)	\(=\)	\( -1008\beta_{9} + 560\beta_{8} + 280\beta_{7} + 1320\beta_{6} - 2692\beta_{5} + 1320\beta_{3} - 2692 \) -1008b9 + 560b8 + 280b7 + 1320b6 - 2692b5 + 1320b3 - 2692
\(\nu^{9}\)	\(=\)	\( -2370\beta_{15} + 1288\beta_{14} - 3722\beta_{13} - 4488\beta_{12} - 3658\beta_{11} - 2370\beta_{10} \) -2370b15 + 1288b14 - 3722b13 - 4488b12 - 3658b11 - 2370b10
\(\nu^{10}\)	\(=\)	\( -10204\beta_{9} + 2888\beta_{8} + 5776\beta_{7} + 13468\beta_{6} - 27064\beta_{5} - 10204\beta_{2} \) -10204b9 + 2888b8 + 5776b7 + 13468b6 - 27064b5 - 10204b2
\(\nu^{11}\)	\(=\)	\( 13092 \beta_{15} - 23924 \beta_{14} - 37768 \beta_{13} - 45804 \beta_{12} - 23924 \beta_{11} + \cdots + 45804 \beta_1 \) 13092b15 - 23924b14 - 37768b13 - 45804b12 - 23924b11 - 37016b10 - 37768b4 + 45804b1
\(\nu^{12}\)	\(=\)	\( -29448\beta_{8} + 29448\beta_{7} - 136944\beta_{3} - 103480\beta_{2} + 273960 \) -29448b8 + 29448b7 - 136944b3 - 103480b2 + 273960
\(\nu^{13}\)	\(=\)	\( 375396\beta_{15} - 375396\beta_{14} + 132928\beta_{11} - 132928\beta_{10} - 383428\beta_{4} + 465712\beta_1 \) 375396b15 - 375396b14 + 132928b11 - 132928b10 - 383428b4 + 465712b1
\(\nu^{14}\)	\(=\)	\( 1050112 \beta_{9} - 598640 \beta_{8} - 299320 \beta_{7} - 1390928 \beta_{6} + 2778808 \beta_{5} + \cdots + 2778808 \) 1050112b9 - 598640b8 - 299320b7 - 1390928b6 + 2778808b5 - 1390928b3 + 2778808
\(\nu^{15}\)	\(=\)	\( 2460264 \beta_{15} - 1349432 \beta_{14} + 3892688 \beta_{13} + 4729928 \beta_{12} + \cdots + 2460264 \beta_{10} \) 2460264b15 - 1349432b14 + 3892688b13 + 4729928b12 + 3809696b11 + 2460264b10

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

\(n\):		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
$3$	\( (T^{8} - 8 T^{6} + 55 T^{4} + \cdots + 81)^{2} \) (T^8 - 8T^6 + 55T^4 - 72*T^2 + 81)^2
$5$	\( (T^{8} - 14 T^{6} + \cdots + 441)^{2} \) (T^8 - 14T^6 + 175T^4 - 294*T^2 + 441)^2
$7$	\( (T^{8} - 14 T^{4} + 2401)^{2} \) (T^8 - 14*T^4 + 2401)^2
$11$	\( (T^{8} - 16 T^{6} + 255 T^{4} + \cdots + 1)^{2} \) (T^8 - 16T^6 + 255T^4 - 16*T^2 + 1)^2
$13$	\( (T^{4} + 56 T^{2} + 756)^{4} \) (T^4 + 56*T^2 + 756)^4
$17$	\( (T^{4} + 4 T^{3} + 19 T^{2} + \cdots + 9)^{4} \) (T^4 + 4T^3 + 19T^2 - 12*T + 9)^4
$19$	\( (T^{8} - 8 T^{6} + 55 T^{4} + \cdots + 81)^{2} \) (T^8 - 8T^6 + 55T^4 - 72*T^2 + 81)^2
$23$	\( (T^{8} + 28 T^{6} + \cdots + 35721)^{2} \) (T^8 + 28T^6 + 595T^4 + 5292*T^2 + 35721)^2
$29$	\( (T^{4} + 56 T^{2} + 84)^{4} \) (T^4 + 56*T^2 + 84)^4
$31$	\( (T^{8} + 28 T^{6} + \cdots + 441)^{2} \) (T^8 + 28T^6 + 763T^4 + 588*T^2 + 441)^2
$37$	\( (T^{8} - 98 T^{6} + \cdots + 2893401)^{2} \) (T^8 - 98T^6 + 7903T^4 - 166698*T^2 + 2893401)^2
$41$	\( (T^{2} + 6 T - 54)^{8} \) (T^2 + 6*T - 54)^8
$43$	\( (T^{4} + 32 T^{2} + 144)^{4} \) (T^4 + 32*T^2 + 144)^4
$47$	\( (T^{8} + 196 T^{6} + \cdots + 85766121)^{2} \) (T^8 + 196T^6 + 29155T^4 + 1815156*T^2 + 85766121)^2
$53$	\( (T^{8} - 98 T^{6} + \cdots + 1058841)^{2} \) (T^8 - 98T^6 + 8575T^4 - 100842*T^2 + 1058841)^2
$59$	\( (T^{8} - 16 T^{6} + 255 T^{4} + \cdots + 1)^{2} \) (T^8 - 16T^6 + 255T^4 - 16*T^2 + 1)^2
$61$	\( (T^{8} - 98 T^{6} + \cdots + 2893401)^{2} \) (T^8 - 98T^6 + 7903T^4 - 166698*T^2 + 2893401)^2
$67$	\( (T^{8} - 144 T^{6} + \cdots + 6561)^{2} \) (T^8 - 144T^6 + 20655T^4 - 11664*T^2 + 6561)^2
$71$	\( (T^{4} - 112 T^{2} + 3024)^{4} \) (T^4 - 112*T^2 + 3024)^4
$73$	\( (T^{4} - 2 T^{3} + \cdots + 729)^{4} \) (T^4 - 2T^3 + 31T^2 + 54*T + 729)^4
$79$	\( (T^{8} + 196 T^{6} + \cdots + 1058841)^{2} \) (T^8 + 196T^6 + 37387T^4 + 201684*T^2 + 1058841)^2
$83$	\( (T^{2} + 128)^{8} \) (T^2 + 128)^8
$89$	\( (T^{2} - 9 T + 81)^{8} \) (T^2 - 9*T + 81)^8
$97$	\( (T^{2} + 2 T - 62)^{8} \) (T^2 + 2*T - 62)^8
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