LMFDB - Newform orbit 2100.2.x.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1357,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("2100.1357");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.x (of order $4$, 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:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} - 8x^{14} + 8x^{12} - 8x^{10} + 212x^{8} + 248x^{6} + 368x^{4} + 32x^{2} + 100 $ x^16 - 8x^14 + 8x^12 - 8x^10 + 212x^8 + 248x^6 + 368x^4 + 32*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{8} q^{3} + (\beta_{10} - \beta_{4} - \beta_{3}) q^{7} - \beta_{5} q^{9}+O(q^{10}) $ q + b8 * q^3 + (b10 - b4 - b3) * q^7 - b5 * q^9	$ q + \beta_{8} q^{3} + (\beta_{10} - \beta_{4} - \beta_{3}) q^{7} - \beta_{5} q^{9} + (\beta_{7} + \beta_{5} - \beta_{2} + 1) q^{11} + (\beta_{15} + \beta_{14} + \cdots - \beta_{4}) q^{13}+ \cdots + ( - \beta_{5} - \beta_1) q^{99}+O(q^{100}) $ q + b8 * q^3 + (b10 - b4 - b3) * q^7 - b5 * q^9 + (b7 + b5 - b2 + 1) * q^11 + (b15 + b14 - 2b11 - b10 - b6 - b4) q^13 + (b15 - b14 + b12 + b10 + b9 + b7 + 2b5 - 1) q^17 + (b12 + b11 + b10 + b6 + b4 - b3) * q^19 + (b14 - b9 - b5 + 1) * q^21 + (-b10 - b7 - b6 - 2b5 - b4 + 2b2 - 2b1 - 1) q^23 + b3 * q^27 + (-2b10 - b9 - b7 - 2b6 - 2b5 + 1) q^29 + (-2b14 - 2b13 - b12 + b11 - 2b10 + b9 + b7 + 2b5 + b3 - 1) * q^31 + (-b13 + b11 + b8) * q^33 + (-b10 - 4b9 - 2b7 - b6 - 3b5 + b4 + 2b2 + 2b1 + 1) q^37 + (b10 - b9 - b7 + b6 + 2b1 + 1) q^39 + (-2b14 + b13 + b10 + b9 - 2b8 + b7 + 2b5 - 2b3 - 1) * q^41 + (-2b5 - 2b2 + 2b1 - 2) q^43 + (2b15 - 2b14 - 2b12 - b10 + 2b9 + 2b7 - b6 + 4b5 - b4 - 2) * q^47 + (2b15 - 2b10 + b9 + 2b8 + b7 - 2b6 + 3b5 - 2b3 - 2b1 - 1) q^49 + (b7 + b5 - b4 - b2 - 1) * q^51 + (b10 - 2b7 + b6 + 2b5 + b4 - b2 + b1 + 4) * q^53 + (b9 + b7 + b5 - b2 - b1) * q^57 + (-2b13 - 2b8 - 2b6 - 2b4 + 2b3) q^59 + (-2b14 + b9 + 4b8 + b7 + 2b5 + 4b3 - 1) * q^61 + (-b13 + b8) * q^63 + (-b10 - 2b9 + 2b7 - b6 - 2b5 + b4 - 2b2 - 2b1 + 4) q^67 + (2b15 + b13 - 2b12 - 2b11 - 2b10 + b9 + b7 - b6 + 2b5 - b4 + 2b3 - 1) * q^69 + (-3b9 - 3b5 + 2b4 + 3b2 - 2) * q^71 + (3b15 + 3b14 + 2b11 + b10 - 4b8 + b6 + b4) * q^73 + (b15 - b14 + b12 + b10 + 2b9 + 2b7 + 2b6 + 2b5 - 4b3 - b2 - b1) q^77 + (4b10 + 2b9 + 2b7 + 4b6 - 2) * q^79 - q^81 + (2b13 + 4b11 + 3b10 - 4b8 + 3b6 + 3b4) * q^83 + (2b15 - 2b14 - b10 + 2b9 + 2b7 + b6 + 4b5 + b4 + 2b3 - 2) * q^87 + (4b15 - 3b12 - 3b11 - 3b10 + 2b9 + 3b8 + 2b7 - 3b6 + 4b5 - 3b4 - 2) * q^89 + (-4b14 - b13 + b12 - b11 - b10 + 3b8 - b4 + 2b3 + 4b2) * q^91 + (-b10 - 2b7 - b6 - b5 - b4 + b2 - b1 + 1) q^93 + (5b15 - 5b14 - 2b12 - 3b10 + 5b9 + 5b7 + b6 + 10b5 + b4 + 2b3 - 5) * q^97 + (-b5 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{11} + 8 q^{21} - 8 q^{23} - 16 q^{37} - 48 q^{43} - 16 q^{51} + 40 q^{53} + 8 q^{57} + 48 q^{67} - 32 q^{71} + 24 q^{77} - 16 q^{81} + 32 q^{91} + 8 q^{93}+O(q^{100}) $ 16 * q + 16 * q^11 + 8 * q^21 - 8 * q^23 - 16 * q^37 - 48 * q^43 - 16 * q^51 + 40 * q^53 + 8 * q^57 + 48 * q^67 - 32 * q^71 + 24 * q^77 - 16 * q^81 + 32 * q^91 + 8 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8x^{14} + 8x^{12} - 8x^{10} + 212x^{8} + 248x^{6} + 368x^{4} + 32x^{2} + 100 $ x^16 - 8*x^14 + 8*x^12 - 8*x^10 + 212*x^8 + 248*x^6 + 368*x^4 + 32*x^2 + 100 :

$\beta_{1}$	$=$	$ ( 168 \nu^{14} + 313 \nu^{12} - 16892 \nu^{10} + 53080 \nu^{8} - 40938 \nu^{6} + 450802 \nu^{4} + \cdots + 148090 ) / 255270 $ (168v^14 + 313v^12 - 16892v^10 + 53080v^8 - 40938v^6 + 450802v^4 - 403288*v^2 + 148090) / 255270
$\beta_{2}$	$=$	$ ( - 926 \nu^{14} + 35451 \nu^{12} - 243201 \nu^{10} + 298601 \nu^{8} - 346280 \nu^{6} + \cdots + 5183600 ) / 1276350 $ (-926v^14 + 35451v^12 - 243201v^10 + 298601v^8 - 346280v^6 + 5518342v^4 + 5075426*v^2 + 5183600) / 1276350
$\beta_{3}$	$=$	$ ( - 570 \nu^{15} + 4712 \nu^{13} - 4682 \nu^{11} - 3835 \nu^{9} - 108472 \nu^{7} - 107896 \nu^{5} + \cdots + 194682 \nu ) / 170180 $ (-570v^15 + 4712v^13 - 4682v^11 - 3835v^9 - 108472v^7 - 107896v^5 + 14152v^3 + 194682v) / 170180
$\beta_{4}$	$=$	$ ( - 5309 \nu^{14} + 52964 \nu^{12} - 133744 \nu^{10} + 152449 \nu^{8} - 925640 \nu^{6} + \cdots - 1778150 ) / 1276350 $ (-5309v^14 + 52964v^12 - 133744v^10 + 152449v^8 - 925640v^6 + 42048v^4 - 235896*v^2 - 1778150) / 1276350
$\beta_{5}$	$=$	$ ( 2329 \nu^{14} - 16984 \nu^{12} + 3874 \nu^{10} + 8131 \nu^{8} + 466620 \nu^{6} + 946682 \nu^{4} + \cdots + 235200 ) / 425450 $ (2329v^14 - 16984v^12 + 3874v^10 + 8131v^8 + 466620v^6 + 946682v^4 + 792956*v^2 + 235200) / 425450
$\beta_{6}$	$=$	$ ( - 11316 \nu^{15} - 8064 \nu^{14} + 98651 \nu^{13} + 44539 \nu^{12} - 144631 \nu^{11} + \cdots + 379600 ) / 2552700 $ (-11316v^15 - 8064v^14 + 98651v^13 + 44539v^12 - 144631v^11 + 130096v^10 + 68966v^9 - 369536v^8 - 2351580v^7 - 1404540v^6 - 1175218v^5 - 5420342v^4 + 166486v^3 - 2935756v^2 + 4605860*v + 379600) / 2552700
$\beta_{7}$	$=$	$ ( 7589 \nu^{14} - 54794 \nu^{12} + 7819 \nu^{10} + 33071 \nu^{8} + 1427600 \nu^{6} + 3316632 \nu^{4} + \cdots + 3909500 ) / 1276350 $ (7589v^14 - 54794v^12 + 7819v^10 + 33071v^8 + 1427600v^6 + 3316632v^4 + 3429726*v^2 + 3909500) / 1276350
$\beta_{8}$	$=$	$ ( 7803 \nu^{15} - 62803 \nu^{13} + 63288 \nu^{11} - 43563 \nu^{9} + 1587570 \nu^{7} + \cdots + 415970 \nu ) / 850900 $ (7803v^15 - 62803v^13 + 63288v^11 - 43563v^9 + 1587570v^7 + 2005314v^5 + 2316332v^3 + 415970v) / 850900
$\beta_{9}$	$=$	$ ( 12236 \nu^{14} - 94911 \nu^{12} + 60231 \nu^{10} + 17089 \nu^{8} + 2540690 \nu^{6} + 3712778 \nu^{4} + \cdots - 942350 ) / 1276350 $ (12236v^14 - 94911v^12 + 60231v^10 + 17089v^8 + 2540690v^6 + 3712778v^4 + 3670474*v^2 - 942350) / 1276350
$\beta_{10}$	$=$	$ ( 11316 \nu^{15} - 18682 \nu^{14} - 98651 \nu^{13} + 150467 \nu^{12} + 144631 \nu^{11} + \cdots - 3176700 ) / 2552700 $ (11316v^15 - 18682v^14 - 98651v^13 + 150467v^12 + 144631v^11 - 137392v^10 - 68966v^9 - 64638v^8 + 2351580v^7 - 3255820v^6 + 1175218v^5 - 5336246v^4 - 166486v^3 - 3407548v^2 - 4605860*v - 3176700) / 2552700
$\beta_{11}$	$=$	$ ( - 16334 \nu^{15} + 18682 \nu^{14} + 101559 \nu^{13} - 150467 \nu^{12} + 102621 \nu^{11} + \cdots + 3176700 ) / 2552700 $ (-16334v^15 + 18682v^14 + 101559v^13 - 150467v^12 + 102621v^11 + 137392v^10 - 54961v^9 + 64638v^8 - 3695480v^7 + 3255820v^6 - 9322322v^5 + 5336246v^4 - 13804966v^3 + 3407548v^2 - 1360210*v + 3176700) / 2552700
$\beta_{12}$	$=$	$ ( - 23638 \nu^{15} + 18682 \nu^{14} + 230578 \nu^{13} - 150467 \nu^{12} - 532523 \nu^{11} + \cdots + 3176700 ) / 2552700 $ (-23638v^15 + 18682v^14 + 230578v^13 - 150467v^12 - 532523v^11 + 137392v^10 + 615878v^9 + 64638v^8 - 5443240v^7 + 3255820v^6 + 2884356v^5 + 5336246v^4 - 445002v^3 + 3407548v^2 + 11472260*v + 3176700) / 2552700
$\beta_{13}$	$=$	$ ( - 29273 \nu^{15} + 18682 \nu^{14} + 228943 \nu^{13} - 150467 \nu^{12} - 185753 \nu^{11} + \cdots + 3176700 ) / 2552700 $ (-29273v^15 + 18682v^14 + 228943v^13 - 150467v^12 - 185753v^11 + 137392v^10 + 126018v^9 + 64638v^8 - 6059090v^7 + 3255820v^6 - 8361154v^5 + 5336246v^4 - 9991622v^3 + 3407548v^2 - 1660020*v + 3176700) / 2552700
$\beta_{14}$	$=$	$ ( 15299 \nu^{15} + 33799 \nu^{14} - 127039 \nu^{13} - 251609 \nu^{12} + 162509 \nu^{11} + \cdots + 3102000 ) / 2552700 $ (15299v^15 + 33799v^14 - 127039v^13 - 251609v^12 + 162509v^11 + 91294v^10 - 174804v^9 + 98946v^8 + 3259370v^7 + 6768010v^6 + 2681062v^5 + 12709502v^4 + 5233886v^3 + 11857936v^2 + 2801520*v + 3102000) / 2552700
$\beta_{15}$	$=$	$ ( 25290 \nu^{15} - 33799 \nu^{14} - 200555 \nu^{13} + 251609 \nu^{12} + 167875 \nu^{11} + \cdots - 3102000 ) / 2552700 $ (25290v^15 - 33799v^14 - 200555v^13 + 251609v^12 + 167875v^11 - 91294v^10 - 20180v^9 - 98946v^8 + 5151300v^7 - 6768010v^6 + 6855310v^5 - 12709502v^4 + 4591250v^3 - 11857936v^2 - 641960*v - 3102000) / 2552700

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

$\nu$	$=$	$ ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{6} + \beta_{4} ) / 2 $ (b15 + b14 + b13 + b6 + b4) / 2
$\nu^{2}$	$=$	$ ( -\beta_{10} + \beta_{9} - 2\beta_{7} - \beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{2} - \beta _1 + 2 ) / 2 $ (-b10 + b9 - 2b7 - b6 - 2b5 - b4 + b2 - b1 + 2) / 2
$\nu^{3}$	$=$	$ ( 3\beta_{13} - \beta_{12} - \beta_{11} - 2\beta_{10} + 3\beta_{8} + 3\beta_{6} + 3\beta_{4} - 4\beta_{3} ) / 2 $ (3b13 - b12 - b11 - 2b10 + 3b8 + 3b6 + 3b4 - 4b3) / 2
$\nu^{4}$	$=$	$ 2\beta_{9} - 6\beta_{7} + \beta_{5} - 3\beta_{4} + 2\beta_{2} + 7 $ 2b9 - 6b7 + b5 - 3b4 + 2b2 + 7
$\nu^{5}$	$=$	$ ( 14 \beta_{15} - 12 \beta_{14} + 12 \beta_{13} - 8 \beta_{12} - 2 \beta_{11} - 17 \beta_{10} + 13 \beta_{9} + \cdots - 13 ) / 2 $ (14b15 - 12b14 + 12b13 - 8b12 - 2b11 - 17b10 + 13b9 + 12b8 + 13b7 + 19b6 + 26b5 + 19b4 - 10*b3 - 13) / 2
$\nu^{6}$	$=$	$ -4\beta_{10} + 7\beta_{9} - 30\beta_{7} - 4\beta_{6} + 8\beta_{5} - 8\beta_{4} + 7\beta_{2} - 4\beta _1 + 51 $ -4b10 + 7b9 - 30b7 - 4b6 + 8b5 - 8b4 + 7b2 - 4b1 + 51
$\nu^{7}$	$=$	$ 41 \beta_{15} - 19 \beta_{14} + 24 \beta_{13} - 23 \beta_{12} - 6 \beta_{11} - 54 \beta_{10} + 30 \beta_{9} + \cdots - 30 $ 41b15 - 19b14 + 24b13 - 23b12 - 6b11 - 54b10 + 30b9 + 7b8 + 30b7 + 49b6 + 60b5 + 49b4 - 37*b3 - 30
$\nu^{8}$	$=$	$ -50\beta_{10} + 16\beta_{9} - 178\beta_{7} - 50\beta_{6} + 40\beta_{5} - 52\beta_{4} + 42\beta_{2} - 36\beta _1 + 258 $ -50b10 + 16b9 - 178b7 - 50b6 + 40b5 - 52b4 + 42b2 - 36b1 + 258
$\nu^{9}$	$=$	$ 155 \beta_{15} - 141 \beta_{14} + 25 \beta_{13} - 122 \beta_{12} - 6 \beta_{11} - 340 \beta_{10} + \cdots - 148 $ 155b15 - 141b14 + 25b13 - 122b12 - 6b11 - 340b10 + 148b9 + 14b8 + 148b7 + 237b6 + 296b5 + 237b4 - 256*b3 - 148
$\nu^{10}$	$=$	$ - 279 \beta_{10} - 29 \beta_{9} - 980 \beta_{7} - 279 \beta_{6} + 422 \beta_{5} - 281 \beta_{4} + \cdots + 1320 $ -279b10 - 29b9 - 980b7 - 279b6 + 422b5 - 281b4 + 211b2 - 219b1 + 1320
$\nu^{11}$	$=$	$ 768 \beta_{15} - 1056 \beta_{14} - 421 \beta_{13} - 709 \beta_{12} + 99 \beta_{11} - 2148 \beta_{10} + \cdots - 912 $ 768b15 - 1056b14 - 421b13 - 709b12 + 99b11 - 2148b10 + 912b9 - 205b8 + 912b7 + 1117b6 + 1824b5 + 1117b4 - 1288*b3 - 912
$\nu^{12}$	$=$	$ - 1752 \beta_{10} - 878 \beta_{9} - 5128 \beta_{7} - 1752 \beta_{6} + 3166 \beta_{5} - 1154 \beta_{4} + \cdots + 6878 $ -1752b10 - 878b9 - 5128b7 - 1752b6 + 3166b5 - 1154b4 + 914b2 - 1394b1 + 6878
$\nu^{13}$	$=$	$ 3562 \beta_{15} - 6356 \beta_{14} - 5216 \beta_{13} - 3856 \beta_{12} + 1118 \beta_{11} - 12839 \beta_{10} + \cdots - 4959 $ 3562b15 - 6356b14 - 5216b13 - 3856b12 + 1118b11 - 12839b10 + 4959b9 - 3304b8 + 4959b7 + 4885b6 + 9918b5 + 4885b4 - 6958*b3 - 4959
$\nu^{14}$	$=$	$ - 11166 \beta_{10} - 8796 \beta_{9} - 26354 \beta_{7} - 11166 \beta_{6} + 20760 \beta_{5} + \cdots + 33556 $ -11166b10 - 8796b9 - 26354b7 - 11166b6 + 20760b5 - 4774b4 + 3782b2 - 8704b1 + 33556
$\nu^{15}$	$=$	$ 13138 \beta_{15} - 37998 \beta_{14} - 45324 \beta_{13} - 20110 \beta_{12} + 9788 \beta_{11} + \cdots - 25568 $ 13138b15 - 37998b14 - 45324b13 - 20110b12 + 9788b11 - 74344b10 + 25568b9 - 28154b8 + 25568b7 + 18698b6 + 51136b5 + 18698b4 - 37578*b3 - 25568

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$-\beta_{5}$	$-1$

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} $

1357.1

0.919224	+	1.30296i
0.550947	−	0.483398i
−0.522506	−	1.01508i
−2.36188	+	0.195512i
2.36188	−	0.195512i
−0.919224	−	1.30296i
−0.550947	+	0.483398i
0.522506	+	1.01508i
0.919224	−	1.30296i
0.550947	+	0.483398i
−0.522506	+	1.01508i
−2.36188	−	0.195512i
2.36188	+	0.195512i
−0.919224	+	1.30296i
−0.550947	−	0.483398i
0.522506	−	1.01508i

−0.707107

−

0.707107i

−2.16604

−

1.51930i

1.00000i

1357.2

−0.707107

−

0.707107i

−1.05782

2.42508i

1.00000i

1357.3

−0.707107

−

0.707107i

0.235858

2.63522i

1.00000i

1357.4

−0.707107

−

0.707107i

1.57379

−

2.12678i

1.00000i

1357.5

0.707107

0.707107i

−2.12678

1.57379i

1.00000i

1357.6

0.707107

0.707107i

−1.51930

−

2.16604i

1.00000i

1357.7

0.707107

0.707107i

2.42508

−

1.05782i

1.00000i

1357.8

0.707107

0.707107i

2.63522

0.235858i

1.00000i

1693.1

−0.707107

0.707107i

−2.16604

1.51930i

−

1.00000i

1693.2

−0.707107

0.707107i

−1.05782

−

2.42508i

−

1.00000i

1693.3

−0.707107

0.707107i

0.235858

−

2.63522i

−

1.00000i

1693.4

−0.707107

0.707107i

1.57379

2.12678i

−

1.00000i

1693.5

0.707107

−

0.707107i

−2.12678

−

1.57379i

−

1.00000i

1693.6

0.707107

−

0.707107i

−1.51930

2.16604i

−

1.00000i

1693.7

0.707107

−

0.707107i

2.42508

1.05782i

−

1.00000i

1693.8

0.707107

−

0.707107i

2.63522

−

0.235858i

−

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
7.b	odd	2	1	inner
35.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2100.2.x.d		16
5.b	even	2	1		420.2.x.a	✓	16
5.c	odd	4	1		420.2.x.a	✓	16
5.c	odd	4	1	inner	2100.2.x.d		16
7.b	odd	2	1	inner	2100.2.x.d		16
15.d	odd	2	1		1260.2.ba.b		16
15.e	even	4	1		1260.2.ba.b		16
20.d	odd	2	1		1680.2.cz.c		16
20.e	even	4	1		1680.2.cz.c		16
35.c	odd	2	1		420.2.x.a	✓	16
35.f	even	4	1		420.2.x.a	✓	16
35.f	even	4	1	inner	2100.2.x.d		16
105.g	even	2	1		1260.2.ba.b		16
105.k	odd	4	1		1260.2.ba.b		16
140.c	even	2	1		1680.2.cz.c		16
140.j	odd	4	1		1680.2.cz.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
420.2.x.a	✓	16	5.b	even	2	1
420.2.x.a	✓	16	5.c	odd	4	1
420.2.x.a	✓	16	35.c	odd	2	1
420.2.x.a	✓	16	35.f	even	4	1
1260.2.ba.b		16	15.d	odd	2	1
1260.2.ba.b		16	15.e	even	4	1
1260.2.ba.b		16	105.g	even	2	1
1260.2.ba.b		16	105.k	odd	4	1
1680.2.cz.c		16	20.d	odd	2	1
1680.2.cz.c		16	20.e	even	4	1
1680.2.cz.c		16	140.c	even	2	1
1680.2.cz.c		16	140.j	odd	4	1
2100.2.x.d		16	1.a	even	1	1	trivial
2100.2.x.d		16	5.c	odd	4	1	inner
2100.2.x.d		16	7.b	odd	2	1	inner
2100.2.x.d		16	35.f	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{4} - 4T_{11}^{3} - 8T_{11}^{2} + 16T_{11} - 4 $ T11^4 - 4*T11^3 - 8*T11^2 + 16*T11 - 4 acting on $S_{2}^{\mathrm{new}}(2100, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{4} + 1)^{4} $ (T^4 + 1)^4
$5$	$ T^{16} $ T^16
$7$	$ T^{16} - 32 T^{13} + \cdots + 5764801 $ T^16 - 32T^13 + 36T^12 - 96T^11 + 512T^10 - 1216T^9 + 3334T^8 - 8512T^7 + 25088T^6 - 32928T^5 + 86436T^4 - 537824*T^3 + 5764801
$11$	$ (T^{4} - 4 T^{3} - 8 T^{2} + \cdots - 4)^{4} $ (T^4 - 4T^3 - 8T^2 + 16*T - 4)^4
$13$	$ T^{16} + \cdots + 116985856 $ T^16 + 1856T^12 + 965760T^8 + 124583936*T^4 + 116985856
$17$	$ T^{16} + 2224 T^{12} + \cdots + 3748096 $ T^16 + 2224T^12 + 1496416T^8 + 313041664*T^4 + 3748096
$19$	$ (T^{8} - 32 T^{6} + \cdots + 16)^{2} $ (T^8 - 32T^6 + 232T^4 - 192*T^2 + 16)^2
$23$	$ (T^{8} + 4 T^{7} + \cdots + 760384)^{2} $ (T^8 + 4T^7 + 8T^6 - 96T^5 + 1856T^4 + 5152T^3 + 10368T^2 - 125568*T + 760384)^2
$29$	$ (T^{8} + 96 T^{6} + \cdots + 30976)^{2} $ (T^8 + 96T^6 + 2080T^4 + 14336*T^2 + 30976)^2
$31$	$ (T^{8} + 112 T^{6} + \cdots + 8464)^{2} $ (T^8 + 112T^6 + 2536T^4 + 13440*T^2 + 8464)^2
$37$	$ (T^{8} + 8 T^{7} + \cdots + 7507600)^{2} $ (T^8 + 8T^7 + 32T^6 - 240T^5 + 9896T^4 + 71328T^3 + 282752T^2 - 2060480*T + 7507600)^2
$41$	$ (T^{8} + 128 T^{6} + \cdots + 270400)^{2} $ (T^8 + 128T^6 + 5088T^4 + 66112*T^2 + 270400)^2
$43$	$ (T^{8} + 24 T^{7} + \cdots + 692224)^{2} $ (T^8 + 24T^7 + 288T^6 + 1472T^5 + 3968T^4 + 4608T^3 + 51200T^2 + 266240*T + 692224)^2
$47$	$ T^{16} + 8704 T^{12} + \cdots + 1048576 $ T^16 + 8704T^12 + 13305856T^8 + 23855104*T^4 + 1048576
$53$	$ (T^{8} - 20 T^{7} + \cdots + 1600)^{2} $ (T^8 - 20T^7 + 200T^6 - 224T^5 + 2784T^4 - 66528T^3 + 798848T^2 - 50560*T + 1600)^2
$59$	$ (T^{8} - 192 T^{6} + \cdots + 4096)^{2} $ (T^8 - 192T^6 + 3712T^4 - 8192*T^2 + 4096)^2
$61$	$ (T^{8} + 176 T^{6} + \cdots + 173056)^{2} $ (T^8 + 176T^6 + 7552T^4 + 86016*T^2 + 173056)^2
$67$	$ (T^{8} - 24 T^{7} + \cdots + 127599616)^{2} $ (T^8 - 24T^7 + 288T^6 - 1376T^5 + 36992T^4 - 781824T^3 + 9056768T^2 - 48075776*T + 127599616)^2
$71$	$ (T^{4} + 8 T^{3} + \cdots + 5164)^{4} $ (T^4 + 8T^3 - 144T^2 - 680*T + 5164)^4
$73$	$ T^{16} + \cdots + 133633600000000 $ T^16 + 52672T^12 + 620167296T^8 + 612807680000*T^4 + 133633600000000
$79$	$ (T^{8} + 384 T^{6} + \cdots + 262144)^{2} $ (T^8 + 384T^6 + 47104T^4 + 1802240*T^2 + 262144)^2
$83$	$ T^{16} + \cdots + 655360000 $ T^16 + 62720T^12 + 234096640T^8 + 61156622336*T^4 + 655360000
$89$	$ (T^{8} - 384 T^{6} + \cdots + 193600)^{2} $ (T^8 - 384T^6 + 25984T^4 - 139328*T^2 + 193600)^2
$97$	$ T^{16} + \cdots + 744753706110976 $ T^16 + 112448T^12 + 1687161984T^8 + 2206901567488*T^4 + 744753706110976
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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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{8} q^{3} + (\beta_{10} - \beta_{4} - \beta_{3}) q^{7} - \beta_{5} q^{9}+O(q^{10}) \) q + b8 * q^3 + (b10 - b4 - b3) * q^7 - b5 * q^9	\( q + \beta_{8} q^{3} + (\beta_{10} - \beta_{4} - \beta_{3}) q^{7} - \beta_{5} q^{9} + (\beta_{7} + \beta_{5} - \beta_{2} + 1) q^{11} + (\beta_{15} + \beta_{14} + \cdots - \beta_{4}) q^{13}+ \cdots + ( - \beta_{5} - \beta_1) q^{99}+O(q^{100}) \) q + b8 * q^3 + (b10 - b4 - b3) * q^7 - b5 * q^9 + (b7 + b5 - b2 + 1) * q^11 + (b15 + b14 - 2b11 - b10 - b6 - b4) q^13 + (b15 - b14 + b12 + b10 + b9 + b7 + 2b5 - 1) q^17 + (b12 + b11 + b10 + b6 + b4 - b3) * q^19 + (b14 - b9 - b5 + 1) * q^21 + (-b10 - b7 - b6 - 2b5 - b4 + 2b2 - 2b1 - 1) q^23 + b3 * q^27 + (-2b10 - b9 - b7 - 2b6 - 2b5 + 1) q^29 + (-2b14 - 2b13 - b12 + b11 - 2b10 + b9 + b7 + 2b5 + b3 - 1) * q^31 + (-b13 + b11 + b8) * q^33 + (-b10 - 4b9 - 2b7 - b6 - 3b5 + b4 + 2b2 + 2b1 + 1) q^37 + (b10 - b9 - b7 + b6 + 2b1 + 1) q^39 + (-2b14 + b13 + b10 + b9 - 2b8 + b7 + 2b5 - 2b3 - 1) * q^41 + (-2b5 - 2b2 + 2b1 - 2) q^43 + (2b15 - 2b14 - 2b12 - b10 + 2b9 + 2b7 - b6 + 4b5 - b4 - 2) * q^47 + (2b15 - 2b10 + b9 + 2b8 + b7 - 2b6 + 3b5 - 2b3 - 2b1 - 1) q^49 + (b7 + b5 - b4 - b2 - 1) * q^51 + (b10 - 2b7 + b6 + 2b5 + b4 - b2 + b1 + 4) * q^53 + (b9 + b7 + b5 - b2 - b1) * q^57 + (-2b13 - 2b8 - 2b6 - 2b4 + 2b3) q^59 + (-2b14 + b9 + 4b8 + b7 + 2b5 + 4b3 - 1) * q^61 + (-b13 + b8) * q^63 + (-b10 - 2b9 + 2b7 - b6 - 2b5 + b4 - 2b2 - 2b1 + 4) q^67 + (2b15 + b13 - 2b12 - 2b11 - 2b10 + b9 + b7 - b6 + 2b5 - b4 + 2b3 - 1) * q^69 + (-3b9 - 3b5 + 2b4 + 3b2 - 2) * q^71 + (3b15 + 3b14 + 2b11 + b10 - 4b8 + b6 + b4) * q^73 + (b15 - b14 + b12 + b10 + 2b9 + 2b7 + 2b6 + 2b5 - 4b3 - b2 - b1) q^77 + (4b10 + 2b9 + 2b7 + 4b6 - 2) * q^79 - q^81 + (2b13 + 4b11 + 3b10 - 4b8 + 3b6 + 3b4) * q^83 + (2b15 - 2b14 - b10 + 2b9 + 2b7 + b6 + 4b5 + b4 + 2b3 - 2) * q^87 + (4b15 - 3b12 - 3b11 - 3b10 + 2b9 + 3b8 + 2b7 - 3b6 + 4b5 - 3b4 - 2) * q^89 + (-4b14 - b13 + b12 - b11 - b10 + 3b8 - b4 + 2b3 + 4b2) * q^91 + (-b10 - 2b7 - b6 - b5 - b4 + b2 - b1 + 1) q^93 + (5b15 - 5b14 - 2b12 - 3b10 + 5b9 + 5b7 + b6 + 10b5 + b4 + 2b3 - 5) * q^97 + (-b5 - b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{11} + 8 q^{21} - 8 q^{23} - 16 q^{37} - 48 q^{43} - 16 q^{51} + 40 q^{53} + 8 q^{57} + 48 q^{67} - 32 q^{71} + 24 q^{77} - 16 q^{81} + 32 q^{91} + 8 q^{93}+O(q^{100}) \) 16 * q + 16 * q^11 + 8 * q^21 - 8 * q^23 - 16 * q^37 - 48 * q^43 - 16 * q^51 + 40 * q^53 + 8 * q^57 + 48 * q^67 - 32 * q^71 + 24 * q^77 - 16 * q^81 + 32 * q^91 + 8 * q^93

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	2100.2.x.d

Level	$2100$
Weight	$2$
Character orbit	2100.x
Analytic conductor	$16.769$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(16.7685844245\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8x^{14} + 8x^{12} - 8x^{10} + 212x^{8} + 248x^{6} + 368x^{4} + 32x^{2} + 100 \) x^16 - 8x^14 + 8x^12 - 8x^10 + 212x^8 + 248x^6 + 368x^4 + 32*x^2 + 100
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{9} \)
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 168 \nu^{14} + 313 \nu^{12} - 16892 \nu^{10} + 53080 \nu^{8} - 40938 \nu^{6} + 450802 \nu^{4} + \cdots + 148090 ) / 255270 \) (168v^14 + 313v^12 - 16892v^10 + 53080v^8 - 40938v^6 + 450802v^4 - 403288*v^2 + 148090) / 255270
\(\beta_{2}\)	\(=\)	\( ( - 926 \nu^{14} + 35451 \nu^{12} - 243201 \nu^{10} + 298601 \nu^{8} - 346280 \nu^{6} + \cdots + 5183600 ) / 1276350 \) (-926v^14 + 35451v^12 - 243201v^10 + 298601v^8 - 346280v^6 + 5518342v^4 + 5075426*v^2 + 5183600) / 1276350
\(\beta_{3}\)	\(=\)	\( ( - 570 \nu^{15} + 4712 \nu^{13} - 4682 \nu^{11} - 3835 \nu^{9} - 108472 \nu^{7} - 107896 \nu^{5} + \cdots + 194682 \nu ) / 170180 \) (-570v^15 + 4712v^13 - 4682v^11 - 3835v^9 - 108472v^7 - 107896v^5 + 14152v^3 + 194682v) / 170180
\(\beta_{4}\)	\(=\)	\( ( - 5309 \nu^{14} + 52964 \nu^{12} - 133744 \nu^{10} + 152449 \nu^{8} - 925640 \nu^{6} + \cdots - 1778150 ) / 1276350 \) (-5309v^14 + 52964v^12 - 133744v^10 + 152449v^8 - 925640v^6 + 42048v^4 - 235896*v^2 - 1778150) / 1276350
\(\beta_{5}\)	\(=\)	\( ( 2329 \nu^{14} - 16984 \nu^{12} + 3874 \nu^{10} + 8131 \nu^{8} + 466620 \nu^{6} + 946682 \nu^{4} + \cdots + 235200 ) / 425450 \) (2329v^14 - 16984v^12 + 3874v^10 + 8131v^8 + 466620v^6 + 946682v^4 + 792956*v^2 + 235200) / 425450
\(\beta_{6}\)	\(=\)	\( ( - 11316 \nu^{15} - 8064 \nu^{14} + 98651 \nu^{13} + 44539 \nu^{12} - 144631 \nu^{11} + \cdots + 379600 ) / 2552700 \) (-11316v^15 - 8064v^14 + 98651v^13 + 44539v^12 - 144631v^11 + 130096v^10 + 68966v^9 - 369536v^8 - 2351580v^7 - 1404540v^6 - 1175218v^5 - 5420342v^4 + 166486v^3 - 2935756v^2 + 4605860*v + 379600) / 2552700
\(\beta_{7}\)	\(=\)	\( ( 7589 \nu^{14} - 54794 \nu^{12} + 7819 \nu^{10} + 33071 \nu^{8} + 1427600 \nu^{6} + 3316632 \nu^{4} + \cdots + 3909500 ) / 1276350 \) (7589v^14 - 54794v^12 + 7819v^10 + 33071v^8 + 1427600v^6 + 3316632v^4 + 3429726*v^2 + 3909500) / 1276350
\(\beta_{8}\)	\(=\)	\( ( 7803 \nu^{15} - 62803 \nu^{13} + 63288 \nu^{11} - 43563 \nu^{9} + 1587570 \nu^{7} + \cdots + 415970 \nu ) / 850900 \) (7803v^15 - 62803v^13 + 63288v^11 - 43563v^9 + 1587570v^7 + 2005314v^5 + 2316332v^3 + 415970v) / 850900
\(\beta_{9}\)	\(=\)	\( ( 12236 \nu^{14} - 94911 \nu^{12} + 60231 \nu^{10} + 17089 \nu^{8} + 2540690 \nu^{6} + 3712778 \nu^{4} + \cdots - 942350 ) / 1276350 \) (12236v^14 - 94911v^12 + 60231v^10 + 17089v^8 + 2540690v^6 + 3712778v^4 + 3670474*v^2 - 942350) / 1276350
\(\beta_{10}\)	\(=\)	\( ( 11316 \nu^{15} - 18682 \nu^{14} - 98651 \nu^{13} + 150467 \nu^{12} + 144631 \nu^{11} + \cdots - 3176700 ) / 2552700 \) (11316v^15 - 18682v^14 - 98651v^13 + 150467v^12 + 144631v^11 - 137392v^10 - 68966v^9 - 64638v^8 + 2351580v^7 - 3255820v^6 + 1175218v^5 - 5336246v^4 - 166486v^3 - 3407548v^2 - 4605860*v - 3176700) / 2552700
\(\beta_{11}\)	\(=\)	\( ( - 16334 \nu^{15} + 18682 \nu^{14} + 101559 \nu^{13} - 150467 \nu^{12} + 102621 \nu^{11} + \cdots + 3176700 ) / 2552700 \) (-16334v^15 + 18682v^14 + 101559v^13 - 150467v^12 + 102621v^11 + 137392v^10 - 54961v^9 + 64638v^8 - 3695480v^7 + 3255820v^6 - 9322322v^5 + 5336246v^4 - 13804966v^3 + 3407548v^2 - 1360210*v + 3176700) / 2552700
\(\beta_{12}\)	\(=\)	\( ( - 23638 \nu^{15} + 18682 \nu^{14} + 230578 \nu^{13} - 150467 \nu^{12} - 532523 \nu^{11} + \cdots + 3176700 ) / 2552700 \) (-23638v^15 + 18682v^14 + 230578v^13 - 150467v^12 - 532523v^11 + 137392v^10 + 615878v^9 + 64638v^8 - 5443240v^7 + 3255820v^6 + 2884356v^5 + 5336246v^4 - 445002v^3 + 3407548v^2 + 11472260*v + 3176700) / 2552700
\(\beta_{13}\)	\(=\)	\( ( - 29273 \nu^{15} + 18682 \nu^{14} + 228943 \nu^{13} - 150467 \nu^{12} - 185753 \nu^{11} + \cdots + 3176700 ) / 2552700 \) (-29273v^15 + 18682v^14 + 228943v^13 - 150467v^12 - 185753v^11 + 137392v^10 + 126018v^9 + 64638v^8 - 6059090v^7 + 3255820v^6 - 8361154v^5 + 5336246v^4 - 9991622v^3 + 3407548v^2 - 1660020*v + 3176700) / 2552700
\(\beta_{14}\)	\(=\)	\( ( 15299 \nu^{15} + 33799 \nu^{14} - 127039 \nu^{13} - 251609 \nu^{12} + 162509 \nu^{11} + \cdots + 3102000 ) / 2552700 \) (15299v^15 + 33799v^14 - 127039v^13 - 251609v^12 + 162509v^11 + 91294v^10 - 174804v^9 + 98946v^8 + 3259370v^7 + 6768010v^6 + 2681062v^5 + 12709502v^4 + 5233886v^3 + 11857936v^2 + 2801520*v + 3102000) / 2552700
\(\beta_{15}\)	\(=\)	\( ( 25290 \nu^{15} - 33799 \nu^{14} - 200555 \nu^{13} + 251609 \nu^{12} + 167875 \nu^{11} + \cdots - 3102000 ) / 2552700 \) (25290v^15 - 33799v^14 - 200555v^13 + 251609v^12 + 167875v^11 - 91294v^10 - 20180v^9 - 98946v^8 + 5151300v^7 - 6768010v^6 + 6855310v^5 - 12709502v^4 + 4591250v^3 - 11857936v^2 - 641960*v - 3102000) / 2552700

\(\nu\)	\(=\)	\( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{6} + \beta_{4} ) / 2 \) (b15 + b14 + b13 + b6 + b4) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{10} + \beta_{9} - 2\beta_{7} - \beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{2} - \beta _1 + 2 ) / 2 \) (-b10 + b9 - 2b7 - b6 - 2b5 - b4 + b2 - b1 + 2) / 2
\(\nu^{3}\)	\(=\)	\( ( 3\beta_{13} - \beta_{12} - \beta_{11} - 2\beta_{10} + 3\beta_{8} + 3\beta_{6} + 3\beta_{4} - 4\beta_{3} ) / 2 \) (3b13 - b12 - b11 - 2b10 + 3b8 + 3b6 + 3b4 - 4b3) / 2
\(\nu^{4}\)	\(=\)	\( 2\beta_{9} - 6\beta_{7} + \beta_{5} - 3\beta_{4} + 2\beta_{2} + 7 \) 2b9 - 6b7 + b5 - 3b4 + 2b2 + 7
\(\nu^{5}\)	\(=\)	\( ( 14 \beta_{15} - 12 \beta_{14} + 12 \beta_{13} - 8 \beta_{12} - 2 \beta_{11} - 17 \beta_{10} + 13 \beta_{9} + \cdots - 13 ) / 2 \) (14b15 - 12b14 + 12b13 - 8b12 - 2b11 - 17b10 + 13b9 + 12b8 + 13b7 + 19b6 + 26b5 + 19b4 - 10*b3 - 13) / 2
\(\nu^{6}\)	\(=\)	\( -4\beta_{10} + 7\beta_{9} - 30\beta_{7} - 4\beta_{6} + 8\beta_{5} - 8\beta_{4} + 7\beta_{2} - 4\beta _1 + 51 \) -4b10 + 7b9 - 30b7 - 4b6 + 8b5 - 8b4 + 7b2 - 4b1 + 51
\(\nu^{7}\)	\(=\)	\( 41 \beta_{15} - 19 \beta_{14} + 24 \beta_{13} - 23 \beta_{12} - 6 \beta_{11} - 54 \beta_{10} + 30 \beta_{9} + \cdots - 30 \) 41b15 - 19b14 + 24b13 - 23b12 - 6b11 - 54b10 + 30b9 + 7b8 + 30b7 + 49b6 + 60b5 + 49b4 - 37*b3 - 30
\(\nu^{8}\)	\(=\)	\( -50\beta_{10} + 16\beta_{9} - 178\beta_{7} - 50\beta_{6} + 40\beta_{5} - 52\beta_{4} + 42\beta_{2} - 36\beta _1 + 258 \) -50b10 + 16b9 - 178b7 - 50b6 + 40b5 - 52b4 + 42b2 - 36b1 + 258
\(\nu^{9}\)	\(=\)	\( 155 \beta_{15} - 141 \beta_{14} + 25 \beta_{13} - 122 \beta_{12} - 6 \beta_{11} - 340 \beta_{10} + \cdots - 148 \) 155b15 - 141b14 + 25b13 - 122b12 - 6b11 - 340b10 + 148b9 + 14b8 + 148b7 + 237b6 + 296b5 + 237b4 - 256*b3 - 148
\(\nu^{10}\)	\(=\)	\( - 279 \beta_{10} - 29 \beta_{9} - 980 \beta_{7} - 279 \beta_{6} + 422 \beta_{5} - 281 \beta_{4} + \cdots + 1320 \) -279b10 - 29b9 - 980b7 - 279b6 + 422b5 - 281b4 + 211b2 - 219b1 + 1320
\(\nu^{11}\)	\(=\)	\( 768 \beta_{15} - 1056 \beta_{14} - 421 \beta_{13} - 709 \beta_{12} + 99 \beta_{11} - 2148 \beta_{10} + \cdots - 912 \) 768b15 - 1056b14 - 421b13 - 709b12 + 99b11 - 2148b10 + 912b9 - 205b8 + 912b7 + 1117b6 + 1824b5 + 1117b4 - 1288*b3 - 912
\(\nu^{12}\)	\(=\)	\( - 1752 \beta_{10} - 878 \beta_{9} - 5128 \beta_{7} - 1752 \beta_{6} + 3166 \beta_{5} - 1154 \beta_{4} + \cdots + 6878 \) -1752b10 - 878b9 - 5128b7 - 1752b6 + 3166b5 - 1154b4 + 914b2 - 1394b1 + 6878
\(\nu^{13}\)	\(=\)	\( 3562 \beta_{15} - 6356 \beta_{14} - 5216 \beta_{13} - 3856 \beta_{12} + 1118 \beta_{11} - 12839 \beta_{10} + \cdots - 4959 \) 3562b15 - 6356b14 - 5216b13 - 3856b12 + 1118b11 - 12839b10 + 4959b9 - 3304b8 + 4959b7 + 4885b6 + 9918b5 + 4885b4 - 6958*b3 - 4959
\(\nu^{14}\)	\(=\)	\( - 11166 \beta_{10} - 8796 \beta_{9} - 26354 \beta_{7} - 11166 \beta_{6} + 20760 \beta_{5} + \cdots + 33556 \) -11166b10 - 8796b9 - 26354b7 - 11166b6 + 20760b5 - 4774b4 + 3782b2 - 8704b1 + 33556
\(\nu^{15}\)	\(=\)	\( 13138 \beta_{15} - 37998 \beta_{14} - 45324 \beta_{13} - 20110 \beta_{12} + 9788 \beta_{11} + \cdots - 25568 \) 13138b15 - 37998b14 - 45324b13 - 20110b12 + 9788b11 - 74344b10 + 25568b9 - 28154b8 + 25568b7 + 18698b6 + 51136b5 + 18698b4 - 37578*b3 - 25568

\(n\)	\(701\)	\(1051\)	\(1177\)	\(1501\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{5}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1357.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{4} + 1)^{4} \) (T^4 + 1)^4
$5$	\( T^{16} \) T^16
$7$	\( T^{16} - 32 T^{13} + \cdots + 5764801 \) T^16 - 32T^13 + 36T^12 - 96T^11 + 512T^10 - 1216T^9 + 3334T^8 - 8512T^7 + 25088T^6 - 32928T^5 + 86436T^4 - 537824*T^3 + 5764801
$11$	\( (T^{4} - 4 T^{3} - 8 T^{2} + \cdots - 4)^{4} \) (T^4 - 4T^3 - 8T^2 + 16*T - 4)^4
$13$	\( T^{16} + \cdots + 116985856 \) T^16 + 1856T^12 + 965760T^8 + 124583936*T^4 + 116985856
$17$	\( T^{16} + 2224 T^{12} + \cdots + 3748096 \) T^16 + 2224T^12 + 1496416T^8 + 313041664*T^4 + 3748096
$19$	\( (T^{8} - 32 T^{6} + \cdots + 16)^{2} \) (T^8 - 32T^6 + 232T^4 - 192*T^2 + 16)^2
$23$	\( (T^{8} + 4 T^{7} + \cdots + 760384)^{2} \) (T^8 + 4T^7 + 8T^6 - 96T^5 + 1856T^4 + 5152T^3 + 10368T^2 - 125568*T + 760384)^2
$29$	\( (T^{8} + 96 T^{6} + \cdots + 30976)^{2} \) (T^8 + 96T^6 + 2080T^4 + 14336*T^2 + 30976)^2
$31$	\( (T^{8} + 112 T^{6} + \cdots + 8464)^{2} \) (T^8 + 112T^6 + 2536T^4 + 13440*T^2 + 8464)^2
$37$	\( (T^{8} + 8 T^{7} + \cdots + 7507600)^{2} \) (T^8 + 8T^7 + 32T^6 - 240T^5 + 9896T^4 + 71328T^3 + 282752T^2 - 2060480*T + 7507600)^2
$41$	\( (T^{8} + 128 T^{6} + \cdots + 270400)^{2} \) (T^8 + 128T^6 + 5088T^4 + 66112*T^2 + 270400)^2
$43$	\( (T^{8} + 24 T^{7} + \cdots + 692224)^{2} \) (T^8 + 24T^7 + 288T^6 + 1472T^5 + 3968T^4 + 4608T^3 + 51200T^2 + 266240*T + 692224)^2
$47$	\( T^{16} + 8704 T^{12} + \cdots + 1048576 \) T^16 + 8704T^12 + 13305856T^8 + 23855104*T^4 + 1048576
$53$	\( (T^{8} - 20 T^{7} + \cdots + 1600)^{2} \) (T^8 - 20T^7 + 200T^6 - 224T^5 + 2784T^4 - 66528T^3 + 798848T^2 - 50560*T + 1600)^2
$59$	\( (T^{8} - 192 T^{6} + \cdots + 4096)^{2} \) (T^8 - 192T^6 + 3712T^4 - 8192*T^2 + 4096)^2
$61$	\( (T^{8} + 176 T^{6} + \cdots + 173056)^{2} \) (T^8 + 176T^6 + 7552T^4 + 86016*T^2 + 173056)^2
$67$	\( (T^{8} - 24 T^{7} + \cdots + 127599616)^{2} \) (T^8 - 24T^7 + 288T^6 - 1376T^5 + 36992T^4 - 781824T^3 + 9056768T^2 - 48075776*T + 127599616)^2
$71$	\( (T^{4} + 8 T^{3} + \cdots + 5164)^{4} \) (T^4 + 8T^3 - 144T^2 - 680*T + 5164)^4
$73$	\( T^{16} + \cdots + 133633600000000 \) T^16 + 52672T^12 + 620167296T^8 + 612807680000*T^4 + 133633600000000
$79$	\( (T^{8} + 384 T^{6} + \cdots + 262144)^{2} \) (T^8 + 384T^6 + 47104T^4 + 1802240*T^2 + 262144)^2
$83$	\( T^{16} + \cdots + 655360000 \) T^16 + 62720T^12 + 234096640T^8 + 61156622336*T^4 + 655360000
$89$	\( (T^{8} - 384 T^{6} + \cdots + 193600)^{2} \) (T^8 - 384T^6 + 25984T^4 - 139328*T^2 + 193600)^2
$97$	\( T^{16} + \cdots + 744753706110976 \) T^16 + 112448T^12 + 1687161984T^8 + 2206901567488*T^4 + 744753706110976
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