LMFDB - Newform orbit 896.2.q.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(703,896)]

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([3, 3, 1]))

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

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

chi := DirichletCharacter("896.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.q (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} + 218x^{12} - 576x^{10} + 1187x^{8} - 576x^{6} + 218x^{4} - 16x^{2} + 1 $ x^16 - 16x^14 + 218x^12 - 576x^10 + 1187x^8 - 576x^6 + 218x^4 - 16*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 7 $
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_{13} q^{3} + \beta_{7} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + \beta_{6} q^{9}+O(q^{10}) $ q + b13 * q^3 + b7 * q^5 + (b14 - b10) * q^7 + b6 * q^9	$ q + \beta_{13} q^{3} + \beta_{7} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + \beta_{6} q^{9} + (\beta_{11} - 2 \beta_{2}) q^{11} - \beta_1 q^{13} + ( - \beta_{15} + \beta_{14} - \beta_{10}) q^{15} + ( - \beta_{6} - \beta_{4} + \beta_{3} - 1) q^{17} + \beta_{2} q^{19} + ( - \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{21}+ \cdots + (\beta_{13} + \beta_{11} + \cdots + \beta_{2}) q^{99}+O(q^{100}) $ q + b13 * q^3 + b7 * q^5 + (b14 - b10) * q^7 + b6 * q^9 + (b11 - 2b2) q^11 - b1 * q^13 + (-b15 + b14 - b10) * q^15 + (-b6 - b4 + b3 - 1) * q^17 + b2 * q^19 + (-b8 + b7 + b5 - b1) * q^21 + (b15 + b14 + b12 - 2b10) q^23 + (2b4 - 2) q^25 + (-2b13 - b11 - 2b9 + b2) * q^27 + (2b8 + b1) q^29 + (-b15 + 2b14 + b12 - b10) q^31 + (-2b6 + b4 - 4b3 - 2) * q^33 + (b13 - 3b11 + 3b9 + 2b2) q^35 + (-b8 - b7 - 2b5 + b1) q^37 + (2b14 - b12 - 2b10) * q^39 + (-2b6 - 4b4 - b3 + 2) * q^41 + (-2b13 + 2b11 + 2b9 + 2b2) * q^43 - b8 * q^45 + (-2b15 + b14 - b12 - 2b10) * q^47 + (4b6 + 2b4 + 5b3 + 1) q^49 + (-4b13 + 2b11 - 2b9 - b2) q^51 + (b8 - b7 + b5 + 2b1) q^53 + (2b15 + 3b14 + b12 - 3b10) q^55 + (2b3 + 1) q^57 + (5b13 - 4b11) * q^59 + (b8 - 3b7 + b1) q^61 - b15 * q^63 + 7b6 q^65 + (-3b13 + 2b11 - 6b9 - 4b2) * q^67 + (b5 - 2b1) q^69 + (-2b6 + 3b4 + 2b3 + 3) q^73 + 2b9 q^75 + (-2b8 - 3b7 - 3b5 - b1) q^77 + (-b15 + b14 + b12 + 2b10) q^79 + (-3b6 - 7b4 - 3b3 + 7) q^81 + (4b13 + 4b11 + 4b9 - 4b2) * q^83 + (2b8 - 2b7 - b5 + b1) * q^85 + (2b15 - 2b14 + b12 - 2b10) q^87 + (-4b6 - b4 - 8b3 + 2) * q^89 + (5b13 - b11 + b9 + 3b2) * q^91 + (-b8 + 3b7 + 6b5 + b1) * q^93 + (-b15 - 2b14 + b10) q^95 + (-6b6 + 8b4 - 3b3 - 4) q^97 + (b13 + b11 - b9 + b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 24 q^{17} - 16 q^{25} - 24 q^{33} + 32 q^{49} + 16 q^{57} + 72 q^{73} + 56 q^{81} + 24 q^{89}+O(q^{100}) $ 16 * q - 24 * q^17 - 16 * q^25 - 24 * q^33 + 32 * q^49 + 16 * q^57 + 72 * q^73 + 56 * q^81 + 24 * q^89

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 16x^{14} + 218x^{12} - 576x^{10} + 1187x^{8} - 576x^{6} + 218x^{4} - 16x^{2} + 1 $ x^16 - 16*x^14 + 218*x^12 - 576*x^10 + 1187*x^8 - 576*x^6 + 218*x^4 - 16*x^2 + 1 :

$\beta_{1}$	$=$	$ ( - 190 \nu^{14} + 3018 \nu^{12} - 40145 \nu^{10} + 92080 \nu^{8} - 45985 \nu^{6} + 17360 \nu^{4} + \cdots + 774670 ) / 207018 $ (-190v^14 + 3018v^12 - 40145v^10 + 92080v^8 - 45985v^6 + 17360v^4 - 1275*v^2 + 774670) / 207018
$\beta_{2}$	$=$	$ ( - 23218 \nu^{15} + 361899 \nu^{13} - 4905719 \nu^{11} + 11252176 \nu^{9} - 21628759 \nu^{7} + \cdots + 3059209 \nu ) / 966084 $ (-23218v^15 + 361899v^13 - 4905719v^11 + 11252176v^9 - 21628759v^7 + 2121392v^5 - 155805v^3 + 3059209v) / 966084
$\beta_{3}$	$=$	$ ( - 2204 \nu^{14} + 34341 \nu^{12} - 465682 \nu^{10} + 1068128 \nu^{8} - 2082722 \nu^{6} + \cdots - 65857 ) / 46746 $ (-2204v^14 + 34341v^12 - 465682v^10 + 1068128v^8 - 2082722v^6 + 201376v^4 - 14790*v^2 - 65857) / 46746
$\beta_{4}$	$=$	$ ( - 18416 \nu^{14} + 293250 \nu^{12} - 3992800 \nu^{10} + 10310543 \nu^{8} - 21178400 \nu^{6} + \cdots + 285221 ) / 241521 $ (-18416v^14 + 293250v^12 - 3992800v^10 + 10310543v^8 - 21178400v^6 + 9233350v^4 - 3886224*v^2 + 285221) / 241521
$\beta_{5}$	$=$	$ ( 38\nu^{14} - 592\nu^{12} + 8029\nu^{10} - 18416\nu^{8} + 36107\nu^{6} - 3472\nu^{4} + 255\nu^{2} + 1144 ) / 434 $ (38v^14 - 592v^12 + 8029v^10 - 18416v^8 + 36107v^6 - 3472v^4 + 255*v^2 + 1144) / 434
$\beta_{6}$	$=$	$ ( - 152822 \nu^{14} + 2463354 \nu^{12} - 33598978 \nu^{10} + 91871363 \nu^{8} - 190220978 \nu^{6} + \cdots + 2567297 ) / 1449126 $ (-152822v^14 + 2463354v^12 - 33598978v^10 + 91871363v^8 - 190220978v^6 + 104838559v^4 - 34978284*v^2 + 2567297) / 1449126
$\beta_{7}$	$=$	$ ( 13657 \nu^{14} - 220032 \nu^{12} + 3000893 \nu^{10} - 8187592 \nu^{8} + 16947499 \nu^{6} + \cdots - 228712 ) / 69006 $ (13657v^14 - 220032v^12 + 3000893v^10 - 8187592v^8 + 16947499v^6 - 9339680v^4 + 3116106*v^2 - 228712) / 69006
$\beta_{8}$	$=$	$ ( - 59053 \nu^{14} + 939930 \nu^{12} - 12797792 \nu^{10} + 32988118 \nu^{8} - 67881376 \nu^{6} + \cdots + 140068 ) / 207018 $ (-59053v^14 + 939930v^12 - 12797792v^10 + 32988118v^8 - 67881376v^6 + 29594894v^4 - 12462339*v^2 + 140068) / 207018
$\beta_{9}$	$=$	$ ( 457406 \nu^{15} - 7125963 \nu^{13} + 96645073 \nu^{11} - 221673392 \nu^{9} + 434493797 \nu^{7} + \cdots + 22085275 \nu ) / 2898252 $ (457406v^15 - 7125963v^13 + 96645073v^11 - 221673392v^9 + 434493797v^7 - 41792464v^5 + 3069435v^3 + 22085275v) / 2898252
$\beta_{10}$	$=$	$ ( 68921 \nu^{15} - 1127190 \nu^{13} + 15418501 \nu^{11} - 45065123 \nu^{9} + 96373433 \nu^{7} + \cdots - 4931417 \nu ) / 414036 $ (68921v^15 - 1127190v^13 + 15418501v^11 - 45065123v^9 + 96373433v^7 - 69366871v^5 + 30279444v^3 - 4931417v) / 414036
$\beta_{11}$	$=$	$ ( - 78207 \nu^{15} + 1241698 \nu^{13} - 16899309 \nu^{11} + 43015895 \nu^{9} - 88172461 \nu^{7} + \cdots + 1186797 \nu ) / 322028 $ (-78207v^15 + 1241698v^13 - 16899309v^11 + 43015895v^9 - 88172461v^7 + 36005291v^5 - 16170710v^3 + 1186797v) / 322028
$\beta_{12}$	$=$	$ ( 30283 \nu^{15} - 488735 \nu^{13} + 6667542 \nu^{11} - 18337048 \nu^{9} + 38055588 \nu^{7} + \cdots - 1186173 \nu ) / 69006 $ (30283v^15 - 488735v^13 + 6667542v^11 - 18337048v^9 + 38055588v^7 - 21670922v^5 + 7536313v^3 - 1186173v) / 69006
$\beta_{13}$	$=$	$ ( 1667753 \nu^{15} - 26751042 \nu^{13} + 364613599 \nu^{11} - 974780855 \nu^{9} + \cdots - 27133613 \nu ) / 2898252 $ (1667753v^15 - 26751042v^13 + 364613599v^11 - 974780855v^9 + 2012090219v^7 - 1024868215v^5 + 369691290v^3 - 27133613v) / 2898252
$\beta_{14}$	$=$	$ ( - 285704 \nu^{15} + 4552695 \nu^{13} - 61980625 \nu^{11} + 160426880 \nu^{9} - 327206033 \nu^{7} + \cdots - 1238167 \nu ) / 414036 $ (-285704v^15 + 4552695v^13 - 61980625v^11 + 160426880v^9 - 327206033v^7 + 139528396v^5 - 45915813v^3 - 1238167v) / 414036
$\beta_{15}$	$=$	$ ( 63811 \nu^{15} - 1011965 \nu^{13} + 13767999 \nu^{11} - 34812769 \nu^{9} + 70853649 \nu^{7} + \cdots + 323670 \nu ) / 69006 $ (63811v^15 - 1011965v^13 + 13767999v^11 - 34812769v^9 + 70853649v^7 - 26853533v^5 + 10367188v^3 + 323670v) / 69006

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

$\nu$	$=$	$ ( 3\beta_{15} + 2\beta_{14} + 2\beta_{12} - 6\beta_{10} - 7\beta_{9} + 7\beta_{2} ) / 14 $ (3b15 + 2b14 + 2b12 - 6b10 - 7b9 + 7b2) / 14
$\nu^{2}$	$=$	$ \beta_{8} + \beta_{7} + 2\beta_{6} + \beta_{5} - 4\beta_{4} + 2\beta_{3} + 4 $ b8 + b7 + 2b6 + b5 - 4b4 + 2*b3 + 4
$\nu^{3}$	$=$	$ ( - 11 \beta_{15} - 26 \beta_{14} - 49 \beta_{13} + 37 \beta_{12} - 91 \beta_{11} - 48 \beta_{10} + \cdots + 91 \beta_{2} ) / 14 $ (-11b15 - 26b14 - 49b13 + 37b12 - 91b11 - 48b10 - 49b9 + 91b2) / 14
$\nu^{4}$	$=$	$ 12\beta_{8} + 16\beta_{7} + 30\beta_{6} - 45\beta_{4} + 12\beta_1 $ 12b8 + 16b7 + 30b6 - 45b4 + 12*b1
$\nu^{5}$	$=$	$ ( -662\beta_{15} - 950\beta_{14} - 539\beta_{13} - 187\beta_{12} - 1239\beta_{11} + 288\beta_{10} ) / 14 $ (-662b15 - 950b14 - 539b13 - 187b12 - 1239b11 + 288b10) / 14
$\nu^{6}$	$=$	$ -217\beta_{5} - 406\beta_{3} + 155\beta _1 - 580 $ -217b5 - 406b3 + 155*b1 - 580
$\nu^{7}$	$=$	$ ( -6241\beta_{15} - 8804\beta_{14} - 8804\beta_{12} + 12482\beta_{10} + 6881\beta_{9} - 16471\beta_{2} ) / 14 $ (-6241b15 - 8804b14 - 8804b12 + 12482b10 + 6881b9 - 16471b2) / 14
$\nu^{8}$	$=$	$ -2040\beta_{8} - 2880\beta_{7} - 5388\beta_{6} - 2880\beta_{5} + 7633\beta_{4} - 5388\beta_{3} - 7633 $ -2040b8 - 2880b7 - 5388b6 - 2880b5 + 7633b4 - 5388b3 - 7633
$\nu^{9}$	$=$	$ ( 34079 \beta_{15} + 48316 \beta_{14} + 90391 \beta_{13} - 82395 \beta_{12} + 217903 \beta_{11} + \cdots - 217903 \beta_{2} ) / 14 $ (34079b15 + 48316b14 + 90391b13 - 82395b12 + 217903b11 + 116474b10 + 90391b9 - 217903b2) / 14
$\nu^{10}$	$=$	$ -26941\beta_{8} - 38089\beta_{7} - 71258\beta_{6} + 100804\beta_{4} - 26941\beta_1 $ -26941b8 - 38089b7 - 71258b6 + 100804b4 - 26941*b1
$\nu^{11}$	$=$	$ ( 1539576 \beta_{15} + 2177450 \beta_{14} + 1193353 \beta_{13} + 450851 \beta_{12} + \cdots - 637874 \beta_{10} ) / 14 $ (1539576b15 + 2177450b14 + 1193353b13 + 450851b12 + 2880283b11 - 637874b10) / 14
$\nu^{12}$	$=$	$ 503440\beta_{5} + 941850\beta_{3} - 356004\beta _1 + 1332045 $ 503440b5 + 941850b3 - 356004*b1 + 1332045
$\nu^{13}$	$=$	$ ( 14387971 \beta_{15} + 20347406 \beta_{14} + 20347406 \beta_{12} - 28775942 \beta_{10} + \cdots + 38066511 \beta_{2} ) / 14 $ (14387971b15 + 20347406b14 + 20347406b12 - 28775942b10 - 15768347b9 + 38066511b2) / 14
$\nu^{14}$	$=$	$ 4704791 \beta_{8} + 6653521 \beta_{7} + 12447598 \beta_{6} + 6653521 \beta_{5} - 17603716 \beta_{4} + \cdots + 17603716 $ 4704791b8 + 6653521b7 + 12447598b6 + 6653521b5 - 17603716b4 + 12447598b3 + 17603716
$\nu^{15}$	$=$	$ ( - 78761323 \beta_{15} - 111386742 \beta_{14} - 208385513 \beta_{13} + 190148065 \beta_{12} + \cdots + 503083399 \beta_{2} ) / 14 $ (-78761323b15 - 111386742b14 - 208385513b13 + 190148065b12 - 503083399b11 - 268909388b10 - 208385513b9 + 503083399b2) / 14

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

Character values

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

$n$	$127$	$129$	$645$
$\chi(n)$	$-1$	$\beta_{4}$	$-1$

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} $

703.1

0.238223	−	0.137538i
−1.30407	+	0.752908i
−3.14831	+	1.81768i
−0.575121	+	0.332046i
3.14831	−	1.81768i
0.575121	−	0.332046i
−0.238223	+	0.137538i
1.30407	−	0.752908i
0.238223	+	0.137538i
−1.30407	−	0.752908i
−3.14831	−	1.81768i
−0.575121	−	0.332046i
3.14831	+	1.81768i
0.575121	+	0.332046i
−0.238223	−	0.137538i
1.30407	+	0.752908i

−1.81952

−

1.05050i

−1.32288

−

2.29129i

1.54230

2.14973i

0.707107

1.22474i

703.2

−1.81952

−

1.05050i

1.32288

2.29129i

−1.54230

−

2.14973i

0.707107

1.22474i

703.3

−1.09057

−

0.629640i

−1.32288

−

2.29129i

−2.57319

0.615370i

−0.707107

−

1.22474i

703.4

−1.09057

−

0.629640i

1.32288

2.29129i

2.57319

−

0.615370i

−0.707107

−

1.22474i

703.5

1.09057

0.629640i

−1.32288

−

2.29129i

2.57319

−

0.615370i

−0.707107

−

1.22474i

703.6

1.09057

0.629640i

1.32288

2.29129i

−2.57319

0.615370i

−0.707107

−

1.22474i

703.7

1.81952

1.05050i

−1.32288

−

2.29129i

−1.54230

−

2.14973i

0.707107

1.22474i

703.8

1.81952

1.05050i

1.32288

2.29129i

1.54230

2.14973i

0.707107

1.22474i

831.1

−1.81952

1.05050i

−1.32288

2.29129i

1.54230

−

2.14973i

0.707107

−

1.22474i

831.2

−1.81952

1.05050i

1.32288

−

2.29129i

−1.54230

2.14973i

0.707107

−

1.22474i

831.3

−1.09057

0.629640i

−1.32288

2.29129i

−2.57319

−

0.615370i

−0.707107

1.22474i

831.4

−1.09057

0.629640i

1.32288

−

2.29129i

2.57319

0.615370i

−0.707107

1.22474i

831.5

1.09057

−

0.629640i

−1.32288

2.29129i

2.57319

0.615370i

−0.707107

1.22474i

831.6

1.09057

−

0.629640i

1.32288

−

2.29129i

−2.57319

−

0.615370i

−0.707107

1.22474i

831.7

1.81952

−

1.05050i

−1.32288

2.29129i

−1.54230

2.14973i

0.707107

−

1.22474i

831.8

1.81952

−

1.05050i

1.32288

−

2.29129i

1.54230

−

2.14973i

0.707107

−

1.22474i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.d	odd	6	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
28.f	even	6	1	inner
56.j	odd	6	1	inner
56.m	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.q.b	✓	16
4.b	odd	2	1	inner	896.2.q.b	✓	16
7.d	odd	6	1	inner	896.2.q.b	✓	16
8.b	even	2	1	inner	896.2.q.b	✓	16
8.d	odd	2	1	inner	896.2.q.b	✓	16
28.f	even	6	1	inner	896.2.q.b	✓	16
56.j	odd	6	1	inner	896.2.q.b	✓	16
56.m	even	6	1	inner	896.2.q.b	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
896.2.q.b	✓	16	1.a	even	1	1	trivial
896.2.q.b	✓	16	4.b	odd	2	1	inner
896.2.q.b	✓	16	7.d	odd	6	1	inner
896.2.q.b	✓	16	8.b	even	2	1	inner
896.2.q.b	✓	16	8.d	odd	2	1	inner
896.2.q.b	✓	16	28.f	even	6	1	inner
896.2.q.b	✓	16	56.j	odd	6	1	inner
896.2.q.b	✓	16	56.m	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 6T_{3}^{6} + 29T_{3}^{4} - 42T_{3}^{2} + 49 $ T3^8 - 6*T3^6 + 29*T3^4 - 42*T3^2 + 49 acting on $S_{2}^{\mathrm{new}}(896, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 6 T^{6} + 29 T^{4} + \cdots + 49)^{2} $ (T^8 - 6T^6 + 29T^4 - 42*T^2 + 49)^2
$5$	$ (T^{4} + 7 T^{2} + 49)^{4} $ (T^4 + 7*T^2 + 49)^4
$7$	$ (T^{8} - 8 T^{6} + \cdots + 2401)^{2} $ (T^8 - 8T^6 + 42T^4 - 392*T^2 + 2401)^2
$11$	$ (T^{8} + 30 T^{6} + \cdots + 3969)^{2} $ (T^8 + 30T^6 + 837T^4 + 1890*T^2 + 3969)^2
$13$	$ (T^{2} - 14)^{8} $ (T^2 - 14)^8
$17$	$ (T^{4} + 6 T^{3} + 9 T^{2} + \cdots + 9)^{4} $ (T^4 + 6T^3 + 9T^2 - 18*T + 9)^4
$19$	$ (T^{8} - 10 T^{6} + \cdots + 49)^{2} $ (T^8 - 10T^6 + 93T^4 - 70*T^2 + 49)^2
$23$	$ (T^{8} - 70 T^{6} + \cdots + 117649)^{2} $ (T^8 - 70T^6 + 4557T^4 - 24010*T^2 + 117649)^2
$29$	$ (T^{2} + 42)^{8} $ (T^2 + 42)^8
$31$	$ (T^{8} + 126 T^{6} + \cdots + 9529569)^{2} $ (T^8 + 126T^6 + 12789T^4 + 388962*T^2 + 9529569)^2
$37$	$ (T^{8} - 126 T^{6} + \cdots + 194481)^{2} $ (T^8 - 126T^6 + 15435T^4 - 55566*T^2 + 194481)^2
$41$	$ (T^{4} + 36 T^{2} + 36)^{4} $ (T^4 + 36*T^2 + 36)^4
$43$	$ (T^{4} - 144 T^{2} + 4032)^{4} $ (T^4 - 144*T^2 + 4032)^4
$47$	$ (T^{8} + 126 T^{6} + \cdots + 9529569)^{2} $ (T^8 + 126T^6 + 12789T^4 + 388962*T^2 + 9529569)^2
$53$	$ (T^{8} - 126 T^{6} + \cdots + 194481)^{2} $ (T^8 - 126T^6 + 15435T^4 - 55566*T^2 + 194481)^2
$59$	$ (T^{8} - 230 T^{6} + \cdots + 45252529)^{2} $ (T^8 - 230T^6 + 46173T^4 - 1547210*T^2 + 45252529)^2
$61$	$ (T^{8} + 154 T^{6} + \cdots + 5764801)^{2} $ (T^8 + 154T^6 + 21315T^4 + 369754*T^2 + 5764801)^2
$67$	$ (T^{8} + 210 T^{6} + \cdots + 9529569)^{2} $ (T^8 + 210T^6 + 41013T^4 + 648270*T^2 + 9529569)^2
$71$	$ T^{16} $ T^16
$73$	$ (T^{4} - 18 T^{3} + 111 T^{2} + \cdots + 9)^{4} $ (T^4 - 18T^3 + 111T^2 - 54*T + 9)^4
$79$	$ (T^{8} - 182 T^{6} + \cdots + 117649)^{2} $ (T^8 - 182T^6 + 32781T^4 - 62426*T^2 + 117649)^2
$83$	$ (T^{4} + 320 T^{2} + 7168)^{4} $ (T^4 + 320*T^2 + 7168)^4
$89$	$ (T^{4} - 6 T^{3} + \cdots + 8649)^{4} $ (T^4 - 6T^3 - 81T^2 + 558*T + 8649)^4
$97$	$ (T^{4} + 204 T^{2} + 36)^{4} $ (T^4 + 204*T^2 + 36)^4
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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.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{13} q^{3} + \beta_{7} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + \beta_{6} q^{9}+O(q^{10}) \) q + b13 * q^3 + b7 * q^5 + (b14 - b10) * q^7 + b6 * q^9	\( q + \beta_{13} q^{3} + \beta_{7} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + \beta_{6} q^{9} + (\beta_{11} - 2 \beta_{2}) q^{11} - \beta_1 q^{13} + ( - \beta_{15} + \beta_{14} - \beta_{10}) q^{15} + ( - \beta_{6} - \beta_{4} + \beta_{3} - 1) q^{17} + \beta_{2} q^{19} + ( - \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{21}+ \cdots + (\beta_{13} + \beta_{11} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) q + b13 * q^3 + b7 * q^5 + (b14 - b10) * q^7 + b6 * q^9 + (b11 - 2b2) q^11 - b1 * q^13 + (-b15 + b14 - b10) * q^15 + (-b6 - b4 + b3 - 1) * q^17 + b2 * q^19 + (-b8 + b7 + b5 - b1) * q^21 + (b15 + b14 + b12 - 2b10) q^23 + (2b4 - 2) q^25 + (-2b13 - b11 - 2b9 + b2) * q^27 + (2b8 + b1) q^29 + (-b15 + 2b14 + b12 - b10) q^31 + (-2b6 + b4 - 4b3 - 2) * q^33 + (b13 - 3b11 + 3b9 + 2b2) q^35 + (-b8 - b7 - 2b5 + b1) q^37 + (2b14 - b12 - 2b10) * q^39 + (-2b6 - 4b4 - b3 + 2) * q^41 + (-2b13 + 2b11 + 2b9 + 2b2) * q^43 - b8 * q^45 + (-2b15 + b14 - b12 - 2b10) * q^47 + (4b6 + 2b4 + 5b3 + 1) q^49 + (-4b13 + 2b11 - 2b9 - b2) q^51 + (b8 - b7 + b5 + 2b1) q^53 + (2b15 + 3b14 + b12 - 3b10) q^55 + (2b3 + 1) q^57 + (5b13 - 4b11) * q^59 + (b8 - 3b7 + b1) q^61 - b15 * q^63 + 7b6 q^65 + (-3b13 + 2b11 - 6b9 - 4b2) * q^67 + (b5 - 2b1) q^69 + (-2b6 + 3b4 + 2b3 + 3) q^73 + 2b9 q^75 + (-2b8 - 3b7 - 3b5 - b1) q^77 + (-b15 + b14 + b12 + 2b10) q^79 + (-3b6 - 7b4 - 3b3 + 7) q^81 + (4b13 + 4b11 + 4b9 - 4b2) * q^83 + (2b8 - 2b7 - b5 + b1) * q^85 + (2b15 - 2b14 + b12 - 2b10) q^87 + (-4b6 - b4 - 8b3 + 2) * q^89 + (5b13 - b11 + b9 + 3b2) * q^91 + (-b8 + 3b7 + 6b5 + b1) * q^93 + (-b15 - 2b14 + b10) q^95 + (-6b6 + 8b4 - 3b3 - 4) q^97 + (b13 + b11 - b9 + b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 24 q^{17} - 16 q^{25} - 24 q^{33} + 32 q^{49} + 16 q^{57} + 72 q^{73} + 56 q^{81} + 24 q^{89}+O(q^{100}) \) 16 * q - 24 * q^17 - 16 * q^25 - 24 * q^33 + 32 * q^49 + 16 * q^57 + 72 * q^73 + 56 * q^81 + 24 * q^89

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.q.b

Level	$896$
Weight	$2$
Character orbit	896.q
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:	\( x^{16} - 16x^{14} + 218x^{12} - 576x^{10} + 1187x^{8} - 576x^{6} + 218x^{4} - 16x^{2} + 1 \) x^16 - 16x^14 + 218x^12 - 576x^10 + 1187x^8 - 576x^6 + 218x^4 - 16*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6}\cdot 7 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 190 \nu^{14} + 3018 \nu^{12} - 40145 \nu^{10} + 92080 \nu^{8} - 45985 \nu^{6} + 17360 \nu^{4} + \cdots + 774670 ) / 207018 \) (-190v^14 + 3018v^12 - 40145v^10 + 92080v^8 - 45985v^6 + 17360v^4 - 1275*v^2 + 774670) / 207018
\(\beta_{2}\)	\(=\)	\( ( - 23218 \nu^{15} + 361899 \nu^{13} - 4905719 \nu^{11} + 11252176 \nu^{9} - 21628759 \nu^{7} + \cdots + 3059209 \nu ) / 966084 \) (-23218v^15 + 361899v^13 - 4905719v^11 + 11252176v^9 - 21628759v^7 + 2121392v^5 - 155805v^3 + 3059209v) / 966084
\(\beta_{3}\)	\(=\)	\( ( - 2204 \nu^{14} + 34341 \nu^{12} - 465682 \nu^{10} + 1068128 \nu^{8} - 2082722 \nu^{6} + \cdots - 65857 ) / 46746 \) (-2204v^14 + 34341v^12 - 465682v^10 + 1068128v^8 - 2082722v^6 + 201376v^4 - 14790*v^2 - 65857) / 46746
\(\beta_{4}\)	\(=\)	\( ( - 18416 \nu^{14} + 293250 \nu^{12} - 3992800 \nu^{10} + 10310543 \nu^{8} - 21178400 \nu^{6} + \cdots + 285221 ) / 241521 \) (-18416v^14 + 293250v^12 - 3992800v^10 + 10310543v^8 - 21178400v^6 + 9233350v^4 - 3886224*v^2 + 285221) / 241521
\(\beta_{5}\)	\(=\)	\( ( 38\nu^{14} - 592\nu^{12} + 8029\nu^{10} - 18416\nu^{8} + 36107\nu^{6} - 3472\nu^{4} + 255\nu^{2} + 1144 ) / 434 \) (38v^14 - 592v^12 + 8029v^10 - 18416v^8 + 36107v^6 - 3472v^4 + 255*v^2 + 1144) / 434
\(\beta_{6}\)	\(=\)	\( ( - 152822 \nu^{14} + 2463354 \nu^{12} - 33598978 \nu^{10} + 91871363 \nu^{8} - 190220978 \nu^{6} + \cdots + 2567297 ) / 1449126 \) (-152822v^14 + 2463354v^12 - 33598978v^10 + 91871363v^8 - 190220978v^6 + 104838559v^4 - 34978284*v^2 + 2567297) / 1449126
\(\beta_{7}\)	\(=\)	\( ( 13657 \nu^{14} - 220032 \nu^{12} + 3000893 \nu^{10} - 8187592 \nu^{8} + 16947499 \nu^{6} + \cdots - 228712 ) / 69006 \) (13657v^14 - 220032v^12 + 3000893v^10 - 8187592v^8 + 16947499v^6 - 9339680v^4 + 3116106*v^2 - 228712) / 69006
\(\beta_{8}\)	\(=\)	\( ( - 59053 \nu^{14} + 939930 \nu^{12} - 12797792 \nu^{10} + 32988118 \nu^{8} - 67881376 \nu^{6} + \cdots + 140068 ) / 207018 \) (-59053v^14 + 939930v^12 - 12797792v^10 + 32988118v^8 - 67881376v^6 + 29594894v^4 - 12462339*v^2 + 140068) / 207018
\(\beta_{9}\)	\(=\)	\( ( 457406 \nu^{15} - 7125963 \nu^{13} + 96645073 \nu^{11} - 221673392 \nu^{9} + 434493797 \nu^{7} + \cdots + 22085275 \nu ) / 2898252 \) (457406v^15 - 7125963v^13 + 96645073v^11 - 221673392v^9 + 434493797v^7 - 41792464v^5 + 3069435v^3 + 22085275v) / 2898252
\(\beta_{10}\)	\(=\)	\( ( 68921 \nu^{15} - 1127190 \nu^{13} + 15418501 \nu^{11} - 45065123 \nu^{9} + 96373433 \nu^{7} + \cdots - 4931417 \nu ) / 414036 \) (68921v^15 - 1127190v^13 + 15418501v^11 - 45065123v^9 + 96373433v^7 - 69366871v^5 + 30279444v^3 - 4931417v) / 414036
\(\beta_{11}\)	\(=\)	\( ( - 78207 \nu^{15} + 1241698 \nu^{13} - 16899309 \nu^{11} + 43015895 \nu^{9} - 88172461 \nu^{7} + \cdots + 1186797 \nu ) / 322028 \) (-78207v^15 + 1241698v^13 - 16899309v^11 + 43015895v^9 - 88172461v^7 + 36005291v^5 - 16170710v^3 + 1186797v) / 322028
\(\beta_{12}\)	\(=\)	\( ( 30283 \nu^{15} - 488735 \nu^{13} + 6667542 \nu^{11} - 18337048 \nu^{9} + 38055588 \nu^{7} + \cdots - 1186173 \nu ) / 69006 \) (30283v^15 - 488735v^13 + 6667542v^11 - 18337048v^9 + 38055588v^7 - 21670922v^5 + 7536313v^3 - 1186173v) / 69006
\(\beta_{13}\)	\(=\)	\( ( 1667753 \nu^{15} - 26751042 \nu^{13} + 364613599 \nu^{11} - 974780855 \nu^{9} + \cdots - 27133613 \nu ) / 2898252 \) (1667753v^15 - 26751042v^13 + 364613599v^11 - 974780855v^9 + 2012090219v^7 - 1024868215v^5 + 369691290v^3 - 27133613v) / 2898252
\(\beta_{14}\)	\(=\)	\( ( - 285704 \nu^{15} + 4552695 \nu^{13} - 61980625 \nu^{11} + 160426880 \nu^{9} - 327206033 \nu^{7} + \cdots - 1238167 \nu ) / 414036 \) (-285704v^15 + 4552695v^13 - 61980625v^11 + 160426880v^9 - 327206033v^7 + 139528396v^5 - 45915813v^3 - 1238167v) / 414036
\(\beta_{15}\)	\(=\)	\( ( 63811 \nu^{15} - 1011965 \nu^{13} + 13767999 \nu^{11} - 34812769 \nu^{9} + 70853649 \nu^{7} + \cdots + 323670 \nu ) / 69006 \) (63811v^15 - 1011965v^13 + 13767999v^11 - 34812769v^9 + 70853649v^7 - 26853533v^5 + 10367188v^3 + 323670v) / 69006

\(\nu\)	\(=\)	\( ( 3\beta_{15} + 2\beta_{14} + 2\beta_{12} - 6\beta_{10} - 7\beta_{9} + 7\beta_{2} ) / 14 \) (3b15 + 2b14 + 2b12 - 6b10 - 7b9 + 7b2) / 14
\(\nu^{2}\)	\(=\)	\( \beta_{8} + \beta_{7} + 2\beta_{6} + \beta_{5} - 4\beta_{4} + 2\beta_{3} + 4 \) b8 + b7 + 2b6 + b5 - 4b4 + 2*b3 + 4
\(\nu^{3}\)	\(=\)	\( ( - 11 \beta_{15} - 26 \beta_{14} - 49 \beta_{13} + 37 \beta_{12} - 91 \beta_{11} - 48 \beta_{10} + \cdots + 91 \beta_{2} ) / 14 \) (-11b15 - 26b14 - 49b13 + 37b12 - 91b11 - 48b10 - 49b9 + 91b2) / 14
\(\nu^{4}\)	\(=\)	\( 12\beta_{8} + 16\beta_{7} + 30\beta_{6} - 45\beta_{4} + 12\beta_1 \) 12b8 + 16b7 + 30b6 - 45b4 + 12*b1
\(\nu^{5}\)	\(=\)	\( ( -662\beta_{15} - 950\beta_{14} - 539\beta_{13} - 187\beta_{12} - 1239\beta_{11} + 288\beta_{10} ) / 14 \) (-662b15 - 950b14 - 539b13 - 187b12 - 1239b11 + 288b10) / 14
\(\nu^{6}\)	\(=\)	\( -217\beta_{5} - 406\beta_{3} + 155\beta _1 - 580 \) -217b5 - 406b3 + 155*b1 - 580
\(\nu^{7}\)	\(=\)	\( ( -6241\beta_{15} - 8804\beta_{14} - 8804\beta_{12} + 12482\beta_{10} + 6881\beta_{9} - 16471\beta_{2} ) / 14 \) (-6241b15 - 8804b14 - 8804b12 + 12482b10 + 6881b9 - 16471b2) / 14
\(\nu^{8}\)	\(=\)	\( -2040\beta_{8} - 2880\beta_{7} - 5388\beta_{6} - 2880\beta_{5} + 7633\beta_{4} - 5388\beta_{3} - 7633 \) -2040b8 - 2880b7 - 5388b6 - 2880b5 + 7633b4 - 5388b3 - 7633
\(\nu^{9}\)	\(=\)	\( ( 34079 \beta_{15} + 48316 \beta_{14} + 90391 \beta_{13} - 82395 \beta_{12} + 217903 \beta_{11} + \cdots - 217903 \beta_{2} ) / 14 \) (34079b15 + 48316b14 + 90391b13 - 82395b12 + 217903b11 + 116474b10 + 90391b9 - 217903b2) / 14
\(\nu^{10}\)	\(=\)	\( -26941\beta_{8} - 38089\beta_{7} - 71258\beta_{6} + 100804\beta_{4} - 26941\beta_1 \) -26941b8 - 38089b7 - 71258b6 + 100804b4 - 26941*b1
\(\nu^{11}\)	\(=\)	\( ( 1539576 \beta_{15} + 2177450 \beta_{14} + 1193353 \beta_{13} + 450851 \beta_{12} + \cdots - 637874 \beta_{10} ) / 14 \) (1539576b15 + 2177450b14 + 1193353b13 + 450851b12 + 2880283b11 - 637874b10) / 14
\(\nu^{12}\)	\(=\)	\( 503440\beta_{5} + 941850\beta_{3} - 356004\beta _1 + 1332045 \) 503440b5 + 941850b3 - 356004*b1 + 1332045
\(\nu^{13}\)	\(=\)	\( ( 14387971 \beta_{15} + 20347406 \beta_{14} + 20347406 \beta_{12} - 28775942 \beta_{10} + \cdots + 38066511 \beta_{2} ) / 14 \) (14387971b15 + 20347406b14 + 20347406b12 - 28775942b10 - 15768347b9 + 38066511b2) / 14
\(\nu^{14}\)	\(=\)	\( 4704791 \beta_{8} + 6653521 \beta_{7} + 12447598 \beta_{6} + 6653521 \beta_{5} - 17603716 \beta_{4} + \cdots + 17603716 \) 4704791b8 + 6653521b7 + 12447598b6 + 6653521b5 - 17603716b4 + 12447598b3 + 17603716
\(\nu^{15}\)	\(=\)	\( ( - 78761323 \beta_{15} - 111386742 \beta_{14} - 208385513 \beta_{13} + 190148065 \beta_{12} + \cdots + 503083399 \beta_{2} ) / 14 \) (-78761323b15 - 111386742b14 - 208385513b13 + 190148065b12 - 503083399b11 - 268909388b10 - 208385513b9 + 503083399b2) / 14

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(-1\)	\(\beta_{4}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 703.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 6 T^{6} + 29 T^{4} + \cdots + 49)^{2} \) (T^8 - 6T^6 + 29T^4 - 42*T^2 + 49)^2
$5$	\( (T^{4} + 7 T^{2} + 49)^{4} \) (T^4 + 7*T^2 + 49)^4
$7$	\( (T^{8} - 8 T^{6} + \cdots + 2401)^{2} \) (T^8 - 8T^6 + 42T^4 - 392*T^2 + 2401)^2
$11$	\( (T^{8} + 30 T^{6} + \cdots + 3969)^{2} \) (T^8 + 30T^6 + 837T^4 + 1890*T^2 + 3969)^2
$13$	\( (T^{2} - 14)^{8} \) (T^2 - 14)^8
$17$	\( (T^{4} + 6 T^{3} + 9 T^{2} + \cdots + 9)^{4} \) (T^4 + 6T^3 + 9T^2 - 18*T + 9)^4
$19$	\( (T^{8} - 10 T^{6} + \cdots + 49)^{2} \) (T^8 - 10T^6 + 93T^4 - 70*T^2 + 49)^2
$23$	\( (T^{8} - 70 T^{6} + \cdots + 117649)^{2} \) (T^8 - 70T^6 + 4557T^4 - 24010*T^2 + 117649)^2
$29$	\( (T^{2} + 42)^{8} \) (T^2 + 42)^8
$31$	\( (T^{8} + 126 T^{6} + \cdots + 9529569)^{2} \) (T^8 + 126T^6 + 12789T^4 + 388962*T^2 + 9529569)^2
$37$	\( (T^{8} - 126 T^{6} + \cdots + 194481)^{2} \) (T^8 - 126T^6 + 15435T^4 - 55566*T^2 + 194481)^2
$41$	\( (T^{4} + 36 T^{2} + 36)^{4} \) (T^4 + 36*T^2 + 36)^4
$43$	\( (T^{4} - 144 T^{2} + 4032)^{4} \) (T^4 - 144*T^2 + 4032)^4
$47$	\( (T^{8} + 126 T^{6} + \cdots + 9529569)^{2} \) (T^8 + 126T^6 + 12789T^4 + 388962*T^2 + 9529569)^2
$53$	\( (T^{8} - 126 T^{6} + \cdots + 194481)^{2} \) (T^8 - 126T^6 + 15435T^4 - 55566*T^2 + 194481)^2
$59$	\( (T^{8} - 230 T^{6} + \cdots + 45252529)^{2} \) (T^8 - 230T^6 + 46173T^4 - 1547210*T^2 + 45252529)^2
$61$	\( (T^{8} + 154 T^{6} + \cdots + 5764801)^{2} \) (T^8 + 154T^6 + 21315T^4 + 369754*T^2 + 5764801)^2
$67$	\( (T^{8} + 210 T^{6} + \cdots + 9529569)^{2} \) (T^8 + 210T^6 + 41013T^4 + 648270*T^2 + 9529569)^2
$71$	\( T^{16} \) T^16
$73$	\( (T^{4} - 18 T^{3} + 111 T^{2} + \cdots + 9)^{4} \) (T^4 - 18T^3 + 111T^2 - 54*T + 9)^4
$79$	\( (T^{8} - 182 T^{6} + \cdots + 117649)^{2} \) (T^8 - 182T^6 + 32781T^4 - 62426*T^2 + 117649)^2
$83$	\( (T^{4} + 320 T^{2} + 7168)^{4} \) (T^4 + 320*T^2 + 7168)^4
$89$	\( (T^{4} - 6 T^{3} + \cdots + 8649)^{4} \) (T^4 - 6T^3 - 81T^2 + 558*T + 8649)^4
$97$	\( (T^{4} + 204 T^{2} + 36)^{4} \) (T^4 + 204*T^2 + 36)^4
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