LMFDB - Newform orbit 1936.2.e.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1936,2,Mod(1935,1936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1936.1935"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1936.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$15.4590378313$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.4526322734619140625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} + 7 x^{14} - 6 x^{13} + 3 x^{12} + 6 x^{11} + 14 x^{10} - 48 x^{9} + 113 x^{8} + \cdots + 256 $ x^16 - 3x^15 + 7x^14 - 6x^13 + 3x^12 + 6x^11 + 14x^10 - 48x^9 + 113x^8 - 96x^7 + 56x^6 + 48x^5 + 48x^4 - 192x^3 + 448x^2 - 384*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 176)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{13} + \beta_{4}) q^{3} + (\beta_{12} + \beta_{5}) q^{5} + ( - \beta_{10} + \beta_1) q^{7} + ( - \beta_{12} - \beta_{9} + \beta_{5}) q^{9} + (\beta_{14} + \beta_{7}) q^{13} + ( - \beta_{13} - \beta_{8} - \beta_{2}) q^{15}+ \cdots + (4 \beta_{9} + 9 \beta_{5} + 4) q^{97}+O(q^{100}) $ q + (-b13 + b4) * q^3 + (b12 + b5) * q^5 + (-b10 + b1) * q^7 + (-b12 - b9 + b5) * q^9 + (b14 + b7) * q^13 + (-b13 - b8 - b2) * q^15 + (b15 - b3) * q^17 + (-2b11 + 2b10 + b6 - b1) * q^19 + (b14 + b7 - 2b3) q^21 - 2b4 q^23 + (-b9 + 3b5 + 3) q^25 + (-4b8 + 2b4 + 3b2) q^27 + (-b15 + b14 - 2b7 - b3) q^29 + (-b13 + 4b4) q^31 + (2b11 - 2b10 - b6 - 2b1) q^35 + (-b9 - b5 + 2) * q^37 + (2b10 - 3b6 - 2b1) q^39 + (-b15 + 3b7) q^41 + (-3b11 + 2b10 - b6 - b1) * q^43 + (b12 - b9 - 4) * q^45 + (2b13 - 3b8 - b2) * q^47 + (-b12 + 2b9 + 5b5 + 5) * q^49 + (-b11 + b10 + 3b6 + b1) q^51 + (b12 - 2b9 + b5 + 2) q^53 + (-b15 - b14 + 3b7 + 2b3) * q^57 + (2b13 + 3b8 - b4 - 4b2) q^59 + (-2b15 - b14 + 3b7) * q^61 + (b10 - 3b6 + b1) q^63 + (-b15 - b14 - 3b3) q^65 + (2b13 + 4b8 + b4 + b2) * q^67 + (-2b5 + 2) q^69 + (b13 - 2b8 + 4b4 + 4b2) q^71 + (b15 + b7 - 4b3) q^73 + (-6b8 + b4 + 2b2) * q^75 + (-3b10 - 3b6 - 3b1) q^79 + (4b9 + 4) q^81 + (-3b11 - b10 - 2b1) * q^83 + (b15 + b14 - 4b7 - 5b3) * q^85 + (-2b11 + 3b10 - 3b6 - 3b1) * q^87 + (-b12 + 2b9 + 4) q^89 + (b13 + 3b8 + 4b4 - 5b2) q^91 + (-b12 - b9 + 4b5 - 6) q^93 + (-2b11 + 6b10 + 5b6 + 2b1) * q^95 + (4b9 + 9b5 + 4) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{5} - 8 q^{9} + 28 q^{25} + 44 q^{37} - 56 q^{45} + 28 q^{49} + 36 q^{53} + 48 q^{69} + 48 q^{81} + 52 q^{89} - 128 q^{93} - 24 q^{97}+O(q^{100}) $ 16 * q - 4 * q^5 - 8 * q^9 + 28 * q^25 + 44 * q^37 - 56 * q^45 + 28 * q^49 + 36 * q^53 + 48 * q^69 + 48 * q^81 + 52 * q^89 - 128 * q^93 - 24 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 3 x^{15} + 7 x^{14} - 6 x^{13} + 3 x^{12} + 6 x^{11} + 14 x^{10} - 48 x^{9} + 113 x^{8} + \cdots + 256 $ x^16 - 3*x^15 + 7*x^14 - 6*x^13 + 3*x^12 + 6*x^11 + 14*x^10 - 48*x^9 + 113*x^8 - 96*x^7 + 56*x^6 + 48*x^5 + 48*x^4 - 192*x^3 + 448*x^2 - 384*x + 256 :

$\beta_{1}$	$=$	$ ( \nu^{14} - 2 \nu^{13} + \nu^{12} + 10 \nu^{11} - 24 \nu^{10} + 35 \nu^{9} - 13 \nu^{8} - 12 \nu^{7} + \cdots + 48 ) / 16 $ (v^14 - 2v^13 + v^12 + 10v^11 - 24v^10 + 35v^9 - 13v^8 - 12v^7 + 23v^6 + 73v^5 - 187v^4 + 280v^3 - 184v^2 + 72v + 48) / 16
$\beta_{2}$	$=$	$ ( 6 \nu^{15} + 105 \nu^{14} - 315 \nu^{13} + 1035 \nu^{12} - 1350 \nu^{11} + 1519 \nu^{10} + \cdots - 19568 ) / 3824 $ (6v^15 + 105v^14 - 315v^13 + 1035v^12 - 1350v^11 + 1519v^10 + 990v^9 - 2220v^8 + 2490v^7 + 6015v^6 - 12228v^5 + 13470v^4 + 4680v^3 - 22800v^2 + 34320*v - 19568) / 3824
$\beta_{3}$	$=$	$ ( - 33 \nu^{15} - 936 \nu^{14} + 2808 \nu^{13} - 6529 \nu^{12} + 5513 \nu^{11} - 2499 \nu^{10} + \cdots - 66368 ) / 15296 $ (-33v^15 - 936v^14 + 2808v^13 - 6529v^12 + 5513v^11 - 2499v^10 - 7118v^9 - 4998v^8 + 22155v^7 - 61643v^6 + 44310v^5 - 27480v^4 - 45816v^3 + 4944v^2 - 3296*v - 66368) / 15296
$\beta_{4}$	$=$	$ ( 85 \nu^{15} - 544 \nu^{14} + 915 \nu^{13} - 992 \nu^{12} - 244 \nu^{11} + 49 \nu^{10} + 1119 \nu^{9} + \cdots - 24192 ) / 7648 $ (85v^15 - 544v^14 + 915v^13 - 992v^12 - 244v^11 + 49v^10 + 1119v^9 - 8028v^8 + 10419v^7 - 10985v^6 - 1867v^5 + 820v^4 + 2248v^3 - 26640v^2 + 25408*v - 24192) / 7648
$\beta_{5}$	$=$	$ ( - \nu^{15} - 9 \nu^{12} + 9 \nu^{11} - 27 \nu^{10} + 18 \nu^{9} - 54 \nu^{8} + 27 \nu^{7} - 99 \nu^{6} + \cdots - 320 ) / 64 $ (-v^15 - 9v^12 + 9v^11 - 27v^10 + 18v^9 - 54v^8 + 27v^7 - 99v^6 + 54v^5 - 216v^4 + 216v^3 - 432v^2 + 288v - 320) / 64
$\beta_{6}$	$=$	$ ( \nu^{15} - 8 \nu^{14} + 12 \nu^{13} - 15 \nu^{12} - 17 \nu^{11} + 31 \nu^{10} - 30 \nu^{9} - 110 \nu^{8} + \cdots - 704 ) / 64 $ (v^15 - 8v^14 + 12v^13 - 15v^12 - 17v^11 + 31v^10 - 30v^9 - 110v^8 + 181v^7 - 221v^6 - 130v^5 + 276v^4 - 296v^3 - 400v^2 + 640v - 704) / 64
$\beta_{7}$	$=$	$ ( 289 \nu^{15} + 158 \nu^{14} - 474 \nu^{13} + 2411 \nu^{12} - 973 \nu^{11} + 549 \nu^{10} + \cdots + 21760 ) / 15296 $ (289v^15 + 158v^14 - 474v^13 + 2411v^12 - 973v^11 + 549v^10 + 6338v^9 + 1098v^8 - 3867v^7 + 22401v^6 - 7734v^5 + 5656v^4 + 19880v^3 + 1200v^2 - 800*v + 21760) / 15296
$\beta_{8}$	$=$	$ ( - 429 \nu^{15} + 1216 \nu^{14} - 2214 \nu^{13} + 2597 \nu^{12} - 509 \nu^{11} + 973 \nu^{10} + \cdots + 39680 ) / 15296 $ (-429v^15 + 1216v^14 - 2214v^13 + 2597v^12 - 509v^11 + 973v^10 - 4104v^9 + 18198v^8 - 24597v^7 + 26537v^6 - 3784v^5 + 7952v^4 - 18184v^3 + 64272v^2 - 58144*v + 39680) / 15296
$\beta_{9}$	$=$	$ ( - \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 2 \nu^{12} + 14 \nu^{11} - 21 \nu^{10} + 15 \nu^{9} + \cdots + 128 ) / 32 $ (-v^15 + 4v^14 - 5v^13 + 2v^12 + 14v^11 - 21v^10 + 15v^9 + 40v^8 - 51v^7 + 41v^6 + 109v^5 - 148v^4 + 184v^3 + 16v^2 - 64v + 128) / 32
$\beta_{10}$	$=$	$ ( - 2 \nu^{15} + 7 \nu^{14} - 13 \nu^{13} + 11 \nu^{12} + 12 \nu^{11} - 29 \nu^{10} + 22 \nu^{9} + \cdots + 768 ) / 64 $ (-2v^15 + 7v^14 - 13v^13 + 11v^12 + 12v^11 - 29v^10 + 22v^9 + 98v^8 - 170v^7 + 217v^6 + 100v^5 - 228v^4 + 312v^3 + 336v^2 - 512*v + 768) / 64
$\beta_{11}$	$=$	$ ( 3 \nu^{15} - 7 \nu^{14} + 13 \nu^{13} - 2 \nu^{12} - 5 \nu^{11} + 8 \nu^{10} + 72 \nu^{9} - 140 \nu^{8} + \cdots - 192 ) / 64 $ (3v^15 - 7v^14 + 13v^13 - 2v^12 - 5v^11 + 8v^10 + 72v^9 - 140v^8 + 191v^7 - 22v^6 - 58v^5 + 60v^4 + 384v^3 - 672v^2 + 736*v - 192) / 64
$\beta_{12}$	$=$	$ ( - 3 \nu^{15} + 10 \nu^{14} - 20 \nu^{13} + 17 \nu^{12} - 7 \nu^{11} + 13 \nu^{10} - 104 \nu^{9} + \cdots + 832 ) / 64 $ (-3v^15 + 10v^14 - 20v^13 + 17v^12 - 7v^11 + 13v^10 - 104v^9 + 242v^8 - 331v^7 + 217v^6 - 52v^5 + 108v^4 - 648v^3 + 1296v^2 - 1344*v + 832) / 64
$\beta_{13}$	$=$	$ ( 410 \nu^{15} - 951 \nu^{14} + 2136 \nu^{13} - 1214 \nu^{12} + 1199 \nu^{11} + 869 \nu^{10} + \cdots - 5120 ) / 7648 $ (410v^15 - 951v^14 + 2136v^13 - 1214v^12 + 1199v^11 + 869v^10 + 7183v^9 - 11646v^8 + 24360v^7 - 10093v^6 + 8807v^5 + 4602v^4 + 33000v^3 - 36048v^2 + 50800*v - 5120) / 7648
$\beta_{14}$	$=$	$ ( 1002 \nu^{15} - 3975 \nu^{14} + 10013 \nu^{13} - 14531 \nu^{12} + 14028 \nu^{11} - 5403 \nu^{10} + \cdots - 90112 ) / 15296 $ (1002v^15 - 3975v^14 + 10013v^13 - 14531v^12 + 14028v^11 - 5403v^10 + 6634v^9 - 30882v^8 + 87922v^7 - 128833v^6 + 124220v^5 - 60684v^4 + 12936v^3 - 21840v^2 + 98688*v - 90112) / 15296
$\beta_{15}$	$=$	$ ( - 1457 \nu^{15} + 2824 \nu^{14} - 3692 \nu^{13} - 3609 \nu^{12} + 8521 \nu^{11} - 5783 \nu^{10} + \cdots + 167424 ) / 15296 $ (-1457v^15 + 2824v^14 - 3692v^13 - 3609v^12 + 8521v^11 - 5783v^10 - 30802v^9 + 60134v^8 - 67861v^7 - 9315v^6 + 67906v^5 - 41836v^4 - 131704v^3 + 299632v^2 - 309376*v + 167424) / 15296

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

$\nu$	$=$	$ ( \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{5} - \beta_{3} + \beta_{2} + 1 ) / 4 $ (b14 - b13 + b12 + b11 + b5 - b3 + b2 + 1) / 4
$\nu^{2}$	$=$	$ ( \beta_{15} + \beta_{14} - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1 ) / 4 $ (b15 + b14 - b11 - b9 - b8 + b7 - b6 + b5 + 2b4 - 2b3 + 2*b2 - 1) / 4
$\nu^{3}$	$=$	$ ( 2\beta_{13} - \beta_{11} - \beta_{10} - \beta_{8} - 4\beta_{7} + \beta_{5} - 3\beta_{4} - 3\beta_{3} - \beta_{2} - 4 ) / 4 $ (2b13 - b11 - b10 - b8 - 4b7 + b5 - 3b4 - 3b3 - b2 - 4) / 4
$\nu^{4}$	$=$	$ ( 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} + \cdots - 3 ) / 4 $ (2b15 - b14 + b13 - b12 + b11 - 2b10 + 2b9 + 2b7 + 2b6 - b5 + 2b4 - 3b3 - 3b2 - 3) / 4
$\nu^{5}$	$=$	$ ( -3\beta_{13} - 3\beta_{12} + \beta_{8} - 4\beta_{5} - \beta_{4} - 2\beta_{2} + 1 ) / 2 $ (-3b13 - 3b12 + b8 - 4b5 - b4 - 2b2 + 1) / 2
$\nu^{6}$	$=$	$ ( - 8 \beta_{13} + 6 \beta_{11} + 4 \beta_{10} - 3 \beta_{8} + \beta_{7} - \beta_{6} - 9 \beta_{5} + \cdots - 17 ) / 4 $ (-8b13 + 6b11 + 4b10 - 3b8 + b7 - b6 - 9b5 + 5b4 + 8b3 + 4b2 + b1 - 17) / 4
$\nu^{7}$	$=$	$ ( - 3 \beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{12} - 10 \beta_{11} + 8 \beta_{10} + \beta_{9} + \cdots + 1 ) / 4 $ (-3b15 - 2b14 + b13 - b12 - 10b11 + 8b10 + b9 - 10b8 - 2b7 - 2b6 - 11b5 + 11b4 + 13b3 + 11*b2 + b1 + 1) / 4
$\nu^{8}$	$=$	$ ( - 13 \beta_{15} - \beta_{14} + 12 \beta_{13} + 12 \beta_{12} - 7 \beta_{11} + 8 \beta_{10} + 11 \beta_{9} + \cdots + 3 ) / 4 $ (-13b15 - b14 + 12b13 + 12b12 - 7b11 + 8b10 + 11b9 - b8 - 9b7 + 9b6 + 13b5 - 10b4 + 6b3 - 2b2 - 4*b1 + 3) / 4
$\nu^{9}$	$=$	$ ( 16 \beta_{13} - 10 \beta_{11} + 8 \beta_{10} + 17 \beta_{8} + 39 \beta_{7} + 9 \beta_{6} + 35 \beta_{5} + \cdots + 31 ) / 4 $ (16b13 - 10b11 + 8b10 + 17b8 + 39b7 + 9b6 + 35b5 + b4 - 4b3 - 8b2 - 9b1 + 31) / 4
$\nu^{10}$	$=$	$ ( 6\beta_{13} - 6\beta_{12} - 15\beta_{9} + 18\beta_{8} + 12\beta_{5} - 33\beta_{4} - 8\beta_{2} + 10 ) / 2 $ (6b13 - 6b12 - 15b9 + 18b8 + 12b5 - 33b4 - 8*b2 + 10) / 2
$\nu^{11}$	$=$	$ ( 27 \beta_{15} - 8 \beta_{14} + 19 \beta_{13} + 19 \beta_{12} + 39 \beta_{11} - 31 \beta_{10} - 27 \beta_{9} + \cdots - 40 ) / 4 $ (27b15 - 8b14 + 19b13 + 19b12 + 39b11 - 31b10 - 27b9 + 28b8 - 3b7 + 3b6 - 9b5 - b4 + 12b3 + 12b2 + 30b1 - 40) / 4
$\nu^{12}$	$=$	$ ( - 6 \beta_{13} - 3 \beta_{11} - 59 \beta_{10} + 31 \beta_{8} - 80 \beta_{7} - 28 \beta_{6} - 63 \beta_{5} + \cdots + 80 ) / 4 $ (-6b13 - 3b11 - 59b10 + 31b8 - 80b7 - 28b6 - 63b5 + 37b4 - 17b3 + 3b2 + 28*b1 + 80) / 4
$\nu^{13}$	$=$	$ ( - 21 \beta_{15} + 26 \beta_{14} - 47 \beta_{13} + 47 \beta_{12} + 82 \beta_{11} - 56 \beta_{10} + \cdots - 15 ) / 4 $ (-21b15 + 26b14 - 47b13 + 47b12 + 82b11 - 56b10 + 73b9 + 6b8 - 62b7 - 62b6 + 53b5 + 67b4 - 61b3 - 21b2 - 41*b1 - 15) / 4
$\nu^{14}$	$=$	$ ( 41 \beta_{15} + 20 \beta_{14} - 61 \beta_{13} - 61 \beta_{12} - 11 \beta_{11} - 9 \beta_{10} + 41 \beta_{9} + \cdots + 92 ) / 4 $ (41b15 + 20b14 - 61b13 - 61b12 - 11b11 - 9b10 + 41b9 - 144b8 + 135b7 - 135b6 + 83b5 + 103b4 - 52b3 + 52b2 - 94*b1 + 92) / 4
$\nu^{15}$	$=$	$ ( -18\beta_{13} - 144\beta_{8} - 20\beta_{5} - 126\beta_{4} + 9\beta_{2} - 91 ) / 2 $ (-18b13 - 144b8 - 20b5 - 126b4 + 9*b2 - 91) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1936\mathbb{Z}\right)^\times$.

$n$	$485$	$849$	$1695$
$\chi(n)$	$1$	$-1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1935.1

0.818796	+	1.15307i
0.0153379	−	1.41413i
1.06903	+	0.925833i
1.21087	−	0.730613i
−1.23234	+	0.693782i
0.589191	−	1.28563i
0.267278	+	1.38873i
−1.23816	+	0.683337i
0.267278	−	1.38873i
−1.23816	−	0.683337i
−1.23234	−	0.693782i
0.589191	+	1.28563i
1.06903	−	0.925833i
1.21087	+	0.730613i
0.818796	−	1.15307i
0.0153379	+	1.41413i

−

3.09898i

0.668269

−2.64994

−6.60365

1935.2

−

3.09898i

0.668269

2.64994

−6.60365

1935.3

−

1.58292i

3.55981

−2.55190

0.494377

1935.4

−

1.58292i

3.55981

2.55190

0.494377

1935.5

−

1.36693i

−2.28630

−0.644621

1.13152

1935.6

−

1.36693i

−2.28630

0.644621

1.13152

1935.7

−

0.149135i

−2.94177

−4.58804

2.97776

1935.8

−

0.149135i

−2.94177

4.58804

2.97776

1935.9

0.149135i

−2.94177

−4.58804

2.97776

1935.10

0.149135i

−2.94177

4.58804

2.97776

1935.11

1.36693i

−2.28630

−0.644621

1.13152

1935.12

1.36693i

−2.28630

0.644621

1.13152

1935.13

1.58292i

3.55981

−2.55190

0.494377

1935.14

1.58292i

3.55981

2.55190

0.494377

1935.15

3.09898i

0.668269

−2.64994

−6.60365

1935.16

3.09898i

0.668269

2.64994

−6.60365

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
11.b	odd	2	1	inner
44.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1936.2.e.f		16
4.b	odd	2	1	inner	1936.2.e.f		16
11.b	odd	2	1	inner	1936.2.e.f		16
11.c	even	5	1		176.2.q.b	✓	16
11.d	odd	10	1		176.2.q.b	✓	16
44.c	even	2	1	inner	1936.2.e.f		16
44.g	even	10	1		176.2.q.b	✓	16
44.h	odd	10	1		176.2.q.b	✓	16
88.k	even	10	1		704.2.u.b		16
88.l	odd	10	1		704.2.u.b		16
88.o	even	10	1		704.2.u.b		16
88.p	odd	10	1		704.2.u.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
176.2.q.b	✓	16	11.c	even	5	1
176.2.q.b	✓	16	11.d	odd	10	1
176.2.q.b	✓	16	44.g	even	10	1
176.2.q.b	✓	16	44.h	odd	10	1
704.2.u.b		16	88.k	even	10	1
704.2.u.b		16	88.l	odd	10	1
704.2.u.b		16	88.o	even	10	1
704.2.u.b		16	88.p	odd	10	1
1936.2.e.f		16	1.a	even	1	1	trivial
1936.2.e.f		16	4.b	odd	2	1	inner
1936.2.e.f		16	11.b	odd	2	1	inner
1936.2.e.f		16	44.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1936, [\chi])$:

$ T_{3}^{8} + 14T_{3}^{6} + 47T_{3}^{4} + 46T_{3}^{2} + 1 $

T3^8 + 14*T3^6 + 47*T3^4 + 46*T3^2 + 1

$ T_{5}^{4} + T_{5}^{3} - 13T_{5}^{2} - 16T_{5} + 16 $

T5^4 + T5^3 - 13*T5^2 - 16*T5 + 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 14 T^{6} + 47 T^{4} + \cdots + 1)^{2} $ (T^8 + 14T^6 + 47T^4 + 46*T^2 + 1)^2
$5$	$ (T^{4} + T^{3} - 13 T^{2} + \cdots + 16)^{4} $ (T^4 + T^3 - 13T^2 - 16T + 16)^4
$7$	$ (T^{8} - 35 T^{6} + \cdots + 400)^{2} $ (T^8 - 35T^6 + 345T^4 - 1100*T^2 + 400)^2
$11$	$ T^{16} $ T^16
$13$	$ (T^{8} + 65 T^{6} + \cdots + 400)^{2} $ (T^8 + 65T^6 + 1265T^4 + 7300*T^2 + 400)^2
$17$	$ (T^{8} + 50 T^{6} + \cdots + 9025)^{2} $ (T^8 + 50T^6 + 755T^4 + 4450*T^2 + 9025)^2
$19$	$ (T^{8} - 90 T^{6} + \cdots + 3025)^{2} $ (T^8 - 90T^6 + 1895T^4 - 4650*T^2 + 3025)^2
$23$	$ (T^{8} + 44 T^{6} + \cdots + 256)^{2} $ (T^8 + 44T^6 + 512T^4 + 1216*T^2 + 256)^2
$29$	$ (T^{8} + 145 T^{6} + \cdots + 336400)^{2} $ (T^8 + 145T^6 + 6305T^4 + 96500*T^2 + 336400)^2
$31$	$ (T^{8} + 149 T^{6} + \cdots + 524176)^{2} $ (T^8 + 149T^6 + 7421T^4 + 133924*T^2 + 524176)^2
$37$	$ (T^{4} - 11 T^{3} + 35 T^{2} + \cdots + 4)^{4} $ (T^4 - 11T^3 + 35T^2 - 28*T + 4)^4
$41$	$ (T^{8} + 170 T^{6} + \cdots + 42025)^{2} $ (T^8 + 170T^6 + 7955T^4 + 100450*T^2 + 42025)^2
$43$	$ (T^{8} - 235 T^{6} + \cdots + 5382400)^{2} $ (T^8 - 235T^6 + 17805T^4 - 530800*T^2 + 5382400)^2
$47$	$ (T^{8} + 181 T^{6} + \cdots + 1752976)^{2} $ (T^8 + 181T^6 + 11261T^4 + 270356*T^2 + 1752976)^2
$53$	$ (T^{4} - 9 T^{3} + \cdots + 396)^{4} $ (T^4 - 9T^3 - 39T^2 + 126*T + 396)^4
$59$	$ (T^{8} + 166 T^{6} + \cdots + 1413721)^{2} $ (T^8 + 166T^6 + 9311T^4 + 202526*T^2 + 1413721)^2
$61$	$ (T^{8} + 385 T^{6} + \cdots + 70224400)^{2} $ (T^8 + 385T^6 + 53705T^4 + 3218900*T^2 + 70224400)^2
$67$	$ (T^{8} + 301 T^{6} + \cdots + 1547536)^{2} $ (T^8 + 301T^6 + 21257T^4 + 402824*T^2 + 1547536)^2
$71$	$ (T^{8} + 317 T^{6} + \cdots + 3083536)^{2} $ (T^8 + 317T^6 + 28829T^4 + 777748*T^2 + 3083536)^2
$73$	$ (T^{8} + 170 T^{6} + \cdots + 156025)^{2} $ (T^8 + 170T^6 + 7715T^4 + 65650*T^2 + 156025)^2
$79$	$ (T^{8} - 495 T^{6} + \cdots + 2624400)^{2} $ (T^8 - 495T^6 + 71685T^4 - 2478600*T^2 + 2624400)^2
$83$	$ (T^{8} - 410 T^{6} + \cdots + 297025)^{2} $ (T^8 - 410T^6 + 35895T^4 - 335450*T^2 + 297025)^2
$89$	$ (T^{4} - 13 T^{3} + \cdots - 164)^{4} $ (T^4 - 13T^3 - T^2 + 328T - 164)^4
$97$	$ (T^{4} + 6 T^{3} + \cdots + 109)^{4} $ (T^4 + 6T^3 - 265T^2 - 402*T + 109)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1936.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{13} + \beta_{4}) q^{3} + (\beta_{12} + \beta_{5}) q^{5} + ( - \beta_{10} + \beta_1) q^{7} + ( - \beta_{12} - \beta_{9} + \beta_{5}) q^{9} + (\beta_{14} + \beta_{7}) q^{13} + ( - \beta_{13} - \beta_{8} - \beta_{2}) q^{15}+ \cdots + (4 \beta_{9} + 9 \beta_{5} + 4) q^{97}+O(q^{100}) \) q + (-b13 + b4) * q^3 + (b12 + b5) * q^5 + (-b10 + b1) * q^7 + (-b12 - b9 + b5) * q^9 + (b14 + b7) * q^13 + (-b13 - b8 - b2) * q^15 + (b15 - b3) * q^17 + (-2b11 + 2b10 + b6 - b1) * q^19 + (b14 + b7 - 2b3) q^21 - 2b4 q^23 + (-b9 + 3b5 + 3) q^25 + (-4b8 + 2b4 + 3b2) q^27 + (-b15 + b14 - 2b7 - b3) q^29 + (-b13 + 4b4) q^31 + (2b11 - 2b10 - b6 - 2b1) q^35 + (-b9 - b5 + 2) * q^37 + (2b10 - 3b6 - 2b1) q^39 + (-b15 + 3b7) q^41 + (-3b11 + 2b10 - b6 - b1) * q^43 + (b12 - b9 - 4) * q^45 + (2b13 - 3b8 - b2) * q^47 + (-b12 + 2b9 + 5b5 + 5) * q^49 + (-b11 + b10 + 3b6 + b1) q^51 + (b12 - 2b9 + b5 + 2) q^53 + (-b15 - b14 + 3b7 + 2b3) * q^57 + (2b13 + 3b8 - b4 - 4b2) q^59 + (-2b15 - b14 + 3b7) * q^61 + (b10 - 3b6 + b1) q^63 + (-b15 - b14 - 3b3) q^65 + (2b13 + 4b8 + b4 + b2) * q^67 + (-2b5 + 2) q^69 + (b13 - 2b8 + 4b4 + 4b2) q^71 + (b15 + b7 - 4b3) q^73 + (-6b8 + b4 + 2b2) * q^75 + (-3b10 - 3b6 - 3b1) q^79 + (4b9 + 4) q^81 + (-3b11 - b10 - 2b1) * q^83 + (b15 + b14 - 4b7 - 5b3) * q^85 + (-2b11 + 3b10 - 3b6 - 3b1) * q^87 + (-b12 + 2b9 + 4) q^89 + (b13 + 3b8 + 4b4 - 5b2) q^91 + (-b12 - b9 + 4b5 - 6) q^93 + (-2b11 + 6b10 + 5b6 + 2b1) * q^95 + (4b9 + 9b5 + 4) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{5} - 8 q^{9} + 28 q^{25} + 44 q^{37} - 56 q^{45} + 28 q^{49} + 36 q^{53} + 48 q^{69} + 48 q^{81} + 52 q^{89} - 128 q^{93} - 24 q^{97}+O(q^{100}) \) 16 * q - 4 * q^5 - 8 * q^9 + 28 * q^25 + 44 * q^37 - 56 * q^45 + 28 * q^49 + 36 * q^53 + 48 * q^69 + 48 * q^81 + 52 * q^89 - 128 * q^93 - 24 * q^97

Introduction

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Modular forms

Varieties

Fields

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	1936.2.e.f

Level	$1936$
Weight	$2$
Character orbit	1936.e
Analytic conductor	$15.459$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(15.4590378313\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.4526322734619140625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 3 x^{15} + 7 x^{14} - 6 x^{13} + 3 x^{12} + 6 x^{11} + 14 x^{10} - 48 x^{9} + 113 x^{8} + \cdots + 256 \) x^16 - 3x^15 + 7x^14 - 6x^13 + 3x^12 + 6x^11 + 14x^10 - 48x^9 + 113x^8 - 96x^7 + 56x^6 + 48x^5 + 48x^4 - 192x^3 + 448x^2 - 384*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	no (minimal twist has level 176)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{14} - 2 \nu^{13} + \nu^{12} + 10 \nu^{11} - 24 \nu^{10} + 35 \nu^{9} - 13 \nu^{8} - 12 \nu^{7} + \cdots + 48 ) / 16 \) (v^14 - 2v^13 + v^12 + 10v^11 - 24v^10 + 35v^9 - 13v^8 - 12v^7 + 23v^6 + 73v^5 - 187v^4 + 280v^3 - 184v^2 + 72v + 48) / 16
\(\beta_{2}\)	\(=\)	\( ( 6 \nu^{15} + 105 \nu^{14} - 315 \nu^{13} + 1035 \nu^{12} - 1350 \nu^{11} + 1519 \nu^{10} + \cdots - 19568 ) / 3824 \) (6v^15 + 105v^14 - 315v^13 + 1035v^12 - 1350v^11 + 1519v^10 + 990v^9 - 2220v^8 + 2490v^7 + 6015v^6 - 12228v^5 + 13470v^4 + 4680v^3 - 22800v^2 + 34320*v - 19568) / 3824
\(\beta_{3}\)	\(=\)	\( ( - 33 \nu^{15} - 936 \nu^{14} + 2808 \nu^{13} - 6529 \nu^{12} + 5513 \nu^{11} - 2499 \nu^{10} + \cdots - 66368 ) / 15296 \) (-33v^15 - 936v^14 + 2808v^13 - 6529v^12 + 5513v^11 - 2499v^10 - 7118v^9 - 4998v^8 + 22155v^7 - 61643v^6 + 44310v^5 - 27480v^4 - 45816v^3 + 4944v^2 - 3296*v - 66368) / 15296
\(\beta_{4}\)	\(=\)	\( ( 85 \nu^{15} - 544 \nu^{14} + 915 \nu^{13} - 992 \nu^{12} - 244 \nu^{11} + 49 \nu^{10} + 1119 \nu^{9} + \cdots - 24192 ) / 7648 \) (85v^15 - 544v^14 + 915v^13 - 992v^12 - 244v^11 + 49v^10 + 1119v^9 - 8028v^8 + 10419v^7 - 10985v^6 - 1867v^5 + 820v^4 + 2248v^3 - 26640v^2 + 25408*v - 24192) / 7648
\(\beta_{5}\)	\(=\)	\( ( - \nu^{15} - 9 \nu^{12} + 9 \nu^{11} - 27 \nu^{10} + 18 \nu^{9} - 54 \nu^{8} + 27 \nu^{7} - 99 \nu^{6} + \cdots - 320 ) / 64 \) (-v^15 - 9v^12 + 9v^11 - 27v^10 + 18v^9 - 54v^8 + 27v^7 - 99v^6 + 54v^5 - 216v^4 + 216v^3 - 432v^2 + 288v - 320) / 64
\(\beta_{6}\)	\(=\)	\( ( \nu^{15} - 8 \nu^{14} + 12 \nu^{13} - 15 \nu^{12} - 17 \nu^{11} + 31 \nu^{10} - 30 \nu^{9} - 110 \nu^{8} + \cdots - 704 ) / 64 \) (v^15 - 8v^14 + 12v^13 - 15v^12 - 17v^11 + 31v^10 - 30v^9 - 110v^8 + 181v^7 - 221v^6 - 130v^5 + 276v^4 - 296v^3 - 400v^2 + 640v - 704) / 64
\(\beta_{7}\)	\(=\)	\( ( 289 \nu^{15} + 158 \nu^{14} - 474 \nu^{13} + 2411 \nu^{12} - 973 \nu^{11} + 549 \nu^{10} + \cdots + 21760 ) / 15296 \) (289v^15 + 158v^14 - 474v^13 + 2411v^12 - 973v^11 + 549v^10 + 6338v^9 + 1098v^8 - 3867v^7 + 22401v^6 - 7734v^5 + 5656v^4 + 19880v^3 + 1200v^2 - 800*v + 21760) / 15296
\(\beta_{8}\)	\(=\)	\( ( - 429 \nu^{15} + 1216 \nu^{14} - 2214 \nu^{13} + 2597 \nu^{12} - 509 \nu^{11} + 973 \nu^{10} + \cdots + 39680 ) / 15296 \) (-429v^15 + 1216v^14 - 2214v^13 + 2597v^12 - 509v^11 + 973v^10 - 4104v^9 + 18198v^8 - 24597v^7 + 26537v^6 - 3784v^5 + 7952v^4 - 18184v^3 + 64272v^2 - 58144*v + 39680) / 15296
\(\beta_{9}\)	\(=\)	\( ( - \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 2 \nu^{12} + 14 \nu^{11} - 21 \nu^{10} + 15 \nu^{9} + \cdots + 128 ) / 32 \) (-v^15 + 4v^14 - 5v^13 + 2v^12 + 14v^11 - 21v^10 + 15v^9 + 40v^8 - 51v^7 + 41v^6 + 109v^5 - 148v^4 + 184v^3 + 16v^2 - 64v + 128) / 32
\(\beta_{10}\)	\(=\)	\( ( - 2 \nu^{15} + 7 \nu^{14} - 13 \nu^{13} + 11 \nu^{12} + 12 \nu^{11} - 29 \nu^{10} + 22 \nu^{9} + \cdots + 768 ) / 64 \) (-2v^15 + 7v^14 - 13v^13 + 11v^12 + 12v^11 - 29v^10 + 22v^9 + 98v^8 - 170v^7 + 217v^6 + 100v^5 - 228v^4 + 312v^3 + 336v^2 - 512*v + 768) / 64
\(\beta_{11}\)	\(=\)	\( ( 3 \nu^{15} - 7 \nu^{14} + 13 \nu^{13} - 2 \nu^{12} - 5 \nu^{11} + 8 \nu^{10} + 72 \nu^{9} - 140 \nu^{8} + \cdots - 192 ) / 64 \) (3v^15 - 7v^14 + 13v^13 - 2v^12 - 5v^11 + 8v^10 + 72v^9 - 140v^8 + 191v^7 - 22v^6 - 58v^5 + 60v^4 + 384v^3 - 672v^2 + 736*v - 192) / 64
\(\beta_{12}\)	\(=\)	\( ( - 3 \nu^{15} + 10 \nu^{14} - 20 \nu^{13} + 17 \nu^{12} - 7 \nu^{11} + 13 \nu^{10} - 104 \nu^{9} + \cdots + 832 ) / 64 \) (-3v^15 + 10v^14 - 20v^13 + 17v^12 - 7v^11 + 13v^10 - 104v^9 + 242v^8 - 331v^7 + 217v^6 - 52v^5 + 108v^4 - 648v^3 + 1296v^2 - 1344*v + 832) / 64
\(\beta_{13}\)	\(=\)	\( ( 410 \nu^{15} - 951 \nu^{14} + 2136 \nu^{13} - 1214 \nu^{12} + 1199 \nu^{11} + 869 \nu^{10} + \cdots - 5120 ) / 7648 \) (410v^15 - 951v^14 + 2136v^13 - 1214v^12 + 1199v^11 + 869v^10 + 7183v^9 - 11646v^8 + 24360v^7 - 10093v^6 + 8807v^5 + 4602v^4 + 33000v^3 - 36048v^2 + 50800*v - 5120) / 7648
\(\beta_{14}\)	\(=\)	\( ( 1002 \nu^{15} - 3975 \nu^{14} + 10013 \nu^{13} - 14531 \nu^{12} + 14028 \nu^{11} - 5403 \nu^{10} + \cdots - 90112 ) / 15296 \) (1002v^15 - 3975v^14 + 10013v^13 - 14531v^12 + 14028v^11 - 5403v^10 + 6634v^9 - 30882v^8 + 87922v^7 - 128833v^6 + 124220v^5 - 60684v^4 + 12936v^3 - 21840v^2 + 98688*v - 90112) / 15296
\(\beta_{15}\)	\(=\)	\( ( - 1457 \nu^{15} + 2824 \nu^{14} - 3692 \nu^{13} - 3609 \nu^{12} + 8521 \nu^{11} - 5783 \nu^{10} + \cdots + 167424 ) / 15296 \) (-1457v^15 + 2824v^14 - 3692v^13 - 3609v^12 + 8521v^11 - 5783v^10 - 30802v^9 + 60134v^8 - 67861v^7 - 9315v^6 + 67906v^5 - 41836v^4 - 131704v^3 + 299632v^2 - 309376*v + 167424) / 15296

\(\nu\)	\(=\)	\( ( \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{5} - \beta_{3} + \beta_{2} + 1 ) / 4 \) (b14 - b13 + b12 + b11 + b5 - b3 + b2 + 1) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} + \beta_{14} - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1 ) / 4 \) (b15 + b14 - b11 - b9 - b8 + b7 - b6 + b5 + 2b4 - 2b3 + 2*b2 - 1) / 4
\(\nu^{3}\)	\(=\)	\( ( 2\beta_{13} - \beta_{11} - \beta_{10} - \beta_{8} - 4\beta_{7} + \beta_{5} - 3\beta_{4} - 3\beta_{3} - \beta_{2} - 4 ) / 4 \) (2b13 - b11 - b10 - b8 - 4b7 + b5 - 3b4 - 3b3 - b2 - 4) / 4
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} + \cdots - 3 ) / 4 \) (2b15 - b14 + b13 - b12 + b11 - 2b10 + 2b9 + 2b7 + 2b6 - b5 + 2b4 - 3b3 - 3b2 - 3) / 4
\(\nu^{5}\)	\(=\)	\( ( -3\beta_{13} - 3\beta_{12} + \beta_{8} - 4\beta_{5} - \beta_{4} - 2\beta_{2} + 1 ) / 2 \) (-3b13 - 3b12 + b8 - 4b5 - b4 - 2b2 + 1) / 2
\(\nu^{6}\)	\(=\)	\( ( - 8 \beta_{13} + 6 \beta_{11} + 4 \beta_{10} - 3 \beta_{8} + \beta_{7} - \beta_{6} - 9 \beta_{5} + \cdots - 17 ) / 4 \) (-8b13 + 6b11 + 4b10 - 3b8 + b7 - b6 - 9b5 + 5b4 + 8b3 + 4b2 + b1 - 17) / 4
\(\nu^{7}\)	\(=\)	\( ( - 3 \beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{12} - 10 \beta_{11} + 8 \beta_{10} + \beta_{9} + \cdots + 1 ) / 4 \) (-3b15 - 2b14 + b13 - b12 - 10b11 + 8b10 + b9 - 10b8 - 2b7 - 2b6 - 11b5 + 11b4 + 13b3 + 11*b2 + b1 + 1) / 4
\(\nu^{8}\)	\(=\)	\( ( - 13 \beta_{15} - \beta_{14} + 12 \beta_{13} + 12 \beta_{12} - 7 \beta_{11} + 8 \beta_{10} + 11 \beta_{9} + \cdots + 3 ) / 4 \) (-13b15 - b14 + 12b13 + 12b12 - 7b11 + 8b10 + 11b9 - b8 - 9b7 + 9b6 + 13b5 - 10b4 + 6b3 - 2b2 - 4*b1 + 3) / 4
\(\nu^{9}\)	\(=\)	\( ( 16 \beta_{13} - 10 \beta_{11} + 8 \beta_{10} + 17 \beta_{8} + 39 \beta_{7} + 9 \beta_{6} + 35 \beta_{5} + \cdots + 31 ) / 4 \) (16b13 - 10b11 + 8b10 + 17b8 + 39b7 + 9b6 + 35b5 + b4 - 4b3 - 8b2 - 9b1 + 31) / 4
\(\nu^{10}\)	\(=\)	\( ( 6\beta_{13} - 6\beta_{12} - 15\beta_{9} + 18\beta_{8} + 12\beta_{5} - 33\beta_{4} - 8\beta_{2} + 10 ) / 2 \) (6b13 - 6b12 - 15b9 + 18b8 + 12b5 - 33b4 - 8*b2 + 10) / 2
\(\nu^{11}\)	\(=\)	\( ( 27 \beta_{15} - 8 \beta_{14} + 19 \beta_{13} + 19 \beta_{12} + 39 \beta_{11} - 31 \beta_{10} - 27 \beta_{9} + \cdots - 40 ) / 4 \) (27b15 - 8b14 + 19b13 + 19b12 + 39b11 - 31b10 - 27b9 + 28b8 - 3b7 + 3b6 - 9b5 - b4 + 12b3 + 12b2 + 30b1 - 40) / 4
\(\nu^{12}\)	\(=\)	\( ( - 6 \beta_{13} - 3 \beta_{11} - 59 \beta_{10} + 31 \beta_{8} - 80 \beta_{7} - 28 \beta_{6} - 63 \beta_{5} + \cdots + 80 ) / 4 \) (-6b13 - 3b11 - 59b10 + 31b8 - 80b7 - 28b6 - 63b5 + 37b4 - 17b3 + 3b2 + 28*b1 + 80) / 4
\(\nu^{13}\)	\(=\)	\( ( - 21 \beta_{15} + 26 \beta_{14} - 47 \beta_{13} + 47 \beta_{12} + 82 \beta_{11} - 56 \beta_{10} + \cdots - 15 ) / 4 \) (-21b15 + 26b14 - 47b13 + 47b12 + 82b11 - 56b10 + 73b9 + 6b8 - 62b7 - 62b6 + 53b5 + 67b4 - 61b3 - 21b2 - 41*b1 - 15) / 4
\(\nu^{14}\)	\(=\)	\( ( 41 \beta_{15} + 20 \beta_{14} - 61 \beta_{13} - 61 \beta_{12} - 11 \beta_{11} - 9 \beta_{10} + 41 \beta_{9} + \cdots + 92 ) / 4 \) (41b15 + 20b14 - 61b13 - 61b12 - 11b11 - 9b10 + 41b9 - 144b8 + 135b7 - 135b6 + 83b5 + 103b4 - 52b3 + 52b2 - 94*b1 + 92) / 4
\(\nu^{15}\)	\(=\)	\( ( -18\beta_{13} - 144\beta_{8} - 20\beta_{5} - 126\beta_{4} + 9\beta_{2} - 91 ) / 2 \) (-18b13 - 144b8 - 20b5 - 126b4 + 9*b2 - 91) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1935.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 14 T^{6} + 47 T^{4} + \cdots + 1)^{2} \) (T^8 + 14T^6 + 47T^4 + 46*T^2 + 1)^2
$5$	\( (T^{4} + T^{3} - 13 T^{2} + \cdots + 16)^{4} \) (T^4 + T^3 - 13T^2 - 16T + 16)^4
$7$	\( (T^{8} - 35 T^{6} + \cdots + 400)^{2} \) (T^8 - 35T^6 + 345T^4 - 1100*T^2 + 400)^2
$11$	\( T^{16} \) T^16
$13$	\( (T^{8} + 65 T^{6} + \cdots + 400)^{2} \) (T^8 + 65T^6 + 1265T^4 + 7300*T^2 + 400)^2
$17$	\( (T^{8} + 50 T^{6} + \cdots + 9025)^{2} \) (T^8 + 50T^6 + 755T^4 + 4450*T^2 + 9025)^2
$19$	\( (T^{8} - 90 T^{6} + \cdots + 3025)^{2} \) (T^8 - 90T^6 + 1895T^4 - 4650*T^2 + 3025)^2
$23$	\( (T^{8} + 44 T^{6} + \cdots + 256)^{2} \) (T^8 + 44T^6 + 512T^4 + 1216*T^2 + 256)^2
$29$	\( (T^{8} + 145 T^{6} + \cdots + 336400)^{2} \) (T^8 + 145T^6 + 6305T^4 + 96500*T^2 + 336400)^2
$31$	\( (T^{8} + 149 T^{6} + \cdots + 524176)^{2} \) (T^8 + 149T^6 + 7421T^4 + 133924*T^2 + 524176)^2
$37$	\( (T^{4} - 11 T^{3} + 35 T^{2} + \cdots + 4)^{4} \) (T^4 - 11T^3 + 35T^2 - 28*T + 4)^4
$41$	\( (T^{8} + 170 T^{6} + \cdots + 42025)^{2} \) (T^8 + 170T^6 + 7955T^4 + 100450*T^2 + 42025)^2
$43$	\( (T^{8} - 235 T^{6} + \cdots + 5382400)^{2} \) (T^8 - 235T^6 + 17805T^4 - 530800*T^2 + 5382400)^2
$47$	\( (T^{8} + 181 T^{6} + \cdots + 1752976)^{2} \) (T^8 + 181T^6 + 11261T^4 + 270356*T^2 + 1752976)^2
$53$	\( (T^{4} - 9 T^{3} + \cdots + 396)^{4} \) (T^4 - 9T^3 - 39T^2 + 126*T + 396)^4
$59$	\( (T^{8} + 166 T^{6} + \cdots + 1413721)^{2} \) (T^8 + 166T^6 + 9311T^4 + 202526*T^2 + 1413721)^2
$61$	\( (T^{8} + 385 T^{6} + \cdots + 70224400)^{2} \) (T^8 + 385T^6 + 53705T^4 + 3218900*T^2 + 70224400)^2
$67$	\( (T^{8} + 301 T^{6} + \cdots + 1547536)^{2} \) (T^8 + 301T^6 + 21257T^4 + 402824*T^2 + 1547536)^2
$71$	\( (T^{8} + 317 T^{6} + \cdots + 3083536)^{2} \) (T^8 + 317T^6 + 28829T^4 + 777748*T^2 + 3083536)^2
$73$	\( (T^{8} + 170 T^{6} + \cdots + 156025)^{2} \) (T^8 + 170T^6 + 7715T^4 + 65650*T^2 + 156025)^2
$79$	\( (T^{8} - 495 T^{6} + \cdots + 2624400)^{2} \) (T^8 - 495T^6 + 71685T^4 - 2478600*T^2 + 2624400)^2
$83$	\( (T^{8} - 410 T^{6} + \cdots + 297025)^{2} \) (T^8 - 410T^6 + 35895T^4 - 335450*T^2 + 297025)^2
$89$	\( (T^{4} - 13 T^{3} + \cdots - 164)^{4} \) (T^4 - 13T^3 - T^2 + 328T - 164)^4
$97$	\( (T^{4} + 6 T^{3} + \cdots + 109)^{4} \) (T^4 + 6T^3 - 265T^2 - 402*T + 109)^4
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\(n\)	\(485\)	\(849\)	\(1695\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)