LMFDB - Newform orbit 405.2.m.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(53,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([10, 9]))

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

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

chi := DirichletCharacter("405.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.m (of order $12$, degree $4$, not 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:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{12})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 100x^{12} - 408x^{10} + 1191x^{8} - 2040x^{6} + 2500x^{4} - 1500x^{2} + 625 $ x^16 - 12x^14 + 100x^12 - 408x^10 + 1191x^8 - 2040x^6 + 2500x^4 - 1500*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-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_{10} + \beta_{9}) q^{2} + ( - \beta_{14} - \beta_{13} - 2 \beta_1) q^{4} + \beta_{7} q^{5} + ( - \beta_{14} - \beta_{8} - \beta_{6} + \cdots - 1) q^{7}+ \cdots + (\beta_{12} + \beta_{10} + \cdots - \beta_{2}) q^{8}+O(q^{10}) $ q + (-b10 + b9) * q^2 + (-b14 - b13 - 2b1) q^4 + b7 * q^5 + (-b14 - b8 - b6 + b5 - b1 - 1) * q^7 + (b12 + b10 - 2b3 - b2) q^8	$ q + ( - \beta_{10} + \beta_{9}) q^{2} + ( - \beta_{14} - \beta_{13} - 2 \beta_1) q^{4} + \beta_{7} q^{5} + ( - \beta_{14} - \beta_{8} - \beta_{6} + \cdots - 1) q^{7}+ \cdots + ( - 2 \beta_{10} + 4 \beta_{3}) q^{98}+O(q^{100}) $ q + (-b10 + b9) * q^2 + (-b14 - b13 - 2b1) q^4 + b7 * q^5 + (-b14 - b8 - b6 + b5 - b1 - 1) * q^7 + (b12 + b10 - 2b3 - b2) q^8 + (-2b13 + 2b11 + b8 + b6 - 2) * q^10 + (-b15 - b10 + b9 - b4 + b2) * q^11 + (-b13 + 2b5 + 2b1) * q^13 + (b12 + b10 - 2b9 - b7 + b4 - b3) q^14 + (2b14 - 2b11 + 2b6 - 2b5 + 2) * q^16 + (-2b15 + b12 - b10 + 2b7 + b2) * q^17 + 3b8 q^19 + (-b15 + b10 - b9 - b4 - 2b2) q^20 + (2b14 + b5 - b1) q^22 - b12 * q^23 + (-2b14 + b11 + 3b8 - 2b6 + b5 + 3b1 - 1) * q^25 + (-b15 - b12 - b10 + b7 - b3 + 2b2) q^26 + (4b13 - 4b11 - 5b8 + 5) q^28 + (b15 + 2b10 - 2b9 - b4 + 2b2) q^29 + (-3b14 + 3b13 - 2b5) q^31 + 2b9 q^32 + (2b14 + 2b11 - 2b8 + 2b6 - 2b1) q^34 + (2b12 - b3 - 2b2) * q^35 + (-2b8 - 3b6 - 2) * q^37 - 3b2 q^38 + (3b14 + b13 - 9b5 + 3b1) q^40 + (3b12 + b10 + 2b9 + b7 + b4 - b3) * q^41 + (2b11 + b8 + b5 + b1 - 1) q^43 + (-b12 + b10 + b2) * q^44 + (-b13 + b11 - b6 - 4) * q^46 + (-4b10 + 4b9) * q^47 + (2b14 + 2b13 - 2b1) q^49 + (-4b12 + b10 - 3b9 - 3b7 + b4 - b3) q^50 + (-2b14 - b8 - 2b6 + b5 - b1 - 1) * q^52 + (-b12 - 2b10 + 4b3 + b2) * q^53 + (b13 - b11 - 3b8 + 2b6 - 4) * q^55 + (2b15 - 3b10 + 3b9 + 2b4 + 3b2) q^56 + (2b13 + 9b5 + 9b1) q^58 + (3b12 - 4b10 + b9 + 4b7 - 4b4 + 4b3) q^59 + (-b5 + 1) * q^61 + (6b15 - 3b12 - b10 - 6b7 - 3b2) * q^62 + (2b13 - 2b11 + 4b8 - 2b6) * q^64 + (2b15 + 2b10 - 2b9 + 3b4 - 3b2) q^65 + (-b14 + 5b5 - 5b1) * q^67 - 2b12 q^68 + (3b14 - 4b11 - 2b8 + 3b6 - 9b5 - 2b1 + 9) * q^70 + (b15 + 3b12 + 3b10 - b7 + b3 - 4b2) q^71 + (-3b13 + 3b11 - b8 + 1) * q^73 + (-3b15 + 5b10 - 5b9 + 3b4 + 5b2) q^74 + (3b14 - 3b13 - 6b5) q^76 + b9 * q^77 + (-b14 - b11 + 3b8 - b6 + 3b1) * q^79 + (-2b12 + 4b10 + 2b3 + 2b2) * q^80 + (11b8 + 8b6 + 11) * q^82 + (-b10 + b9 - 2b4) q^83 + (-3b14 - b13 - b5 + 7b1) * q^85 + (-3b12 - 2b10 - b9 - 2b7 - 2b4 + 2b3) q^86 + (-2b11 + 6b8 + 6b5 + 6b1 - 6) * q^88 + (-3b12 + 3b10 + 3b2) q^89 + (-b13 + b11 - b6 - 1) * q^91 + (-2b15 + 3b10 - 3b9 + b2) q^92 + (-4b14 - 4b13 - 16b1) q^94 + (-3b10 - 3b4 + 3b3) q^95 + (5b14 - 4b8 + 5b6 + 4b5 - 4b1 - 4) q^97 + (-2b10 + 4b3) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{7}+O(q^{10}) $ 16 * q - 8 * q^7	$ 16 q - 8 q^{7} - 32 q^{10} + 16 q^{13} + 16 q^{16} + 8 q^{22} - 8 q^{25} + 80 q^{28} - 16 q^{31} - 32 q^{37} - 72 q^{40} - 8 q^{43} - 64 q^{46} - 8 q^{52} - 64 q^{55} + 72 q^{58} + 8 q^{61} + 40 q^{67} + 72 q^{70} + 16 q^{73} - 48 q^{76} + 176 q^{82} - 8 q^{85} - 48 q^{88} - 16 q^{91} - 32 q^{97}+O(q^{100}) $ 16 * q - 8 * q^7 - 32 * q^10 + 16 * q^13 + 16 * q^16 + 8 * q^22 - 8 * q^25 + 80 * q^28 - 16 * q^31 - 32 * q^37 - 72 * q^40 - 8 * q^43 - 64 * q^46 - 8 * q^52 - 64 * q^55 + 72 * q^58 + 8 * q^61 + 40 * q^67 + 72 * q^70 + 16 * q^73 - 48 * q^76 + 176 * q^82 - 8 * q^85 - 48 * q^88 - 16 * q^91 - 32 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 12x^{14} + 100x^{12} - 408x^{10} + 1191x^{8} - 2040x^{6} + 2500x^{4} - 1500x^{2} + 625 $ x^16 - 12*x^14 + 100*x^12 - 408*x^10 + 1191*x^8 - 2040*x^6 + 2500*x^4 - 1500*x^2 + 625 :

$\beta_{1}$	$=$	$ ( 162 \nu^{14} + 1386 \nu^{12} + 190 \nu^{10} + 31754 \nu^{8} + 610802 \nu^{6} - 1207925 \nu^{4} + \cdots + 1408250 ) / 3170625 $ (162v^14 + 1386v^12 + 190v^10 + 31754v^8 + 610802v^6 - 1207925v^4 + 3359500*v^2 + 1408250) / 3170625
$\beta_{2}$	$=$	$ ( - 2373 \nu^{15} + 25681 \nu^{13} - 196935 \nu^{11} + 583909 \nu^{9} - 895008 \nu^{7} + \cdots - 11788000 \nu ) / 3170625 $ (-2373v^15 + 25681v^13 - 196935v^11 + 583909v^9 - 895008v^7 - 1925650v^5 + 5332125v^3 - 11788000v) / 3170625
$\beta_{3}$	$=$	$ ( 519 \nu^{15} - 11728 \nu^{13} + 111150 \nu^{11} - 674877 \nu^{9} + 2149629 \nu^{7} + \cdots - 3584500 \nu ) / 634125 $ (519v^15 - 11728v^13 + 111150v^11 - 674877v^9 + 2149629v^7 - 4634385v^5 + 4860000v^3 - 3584500v) / 634125
$\beta_{4}$	$=$	$ ( - 3012 \nu^{15} + 20214 \nu^{13} - 103740 \nu^{11} - 410329 \nu^{9} + 3154398 \nu^{7} + \cdots - 14407000 \nu ) / 3170625 $ (-3012v^15 + 20214v^13 - 103740v^11 - 410329v^9 + 3154398v^7 - 12180425v^5 + 18526125v^3 - 14407000v) / 3170625
$\beta_{5}$	$=$	$ ( - 4084 \nu^{14} + 46148 \nu^{12} - 378480 \nu^{10} + 1425447 \nu^{8} - 4067664 \nu^{6} + \cdots + 4996625 ) / 3170625 $ (-4084v^14 + 46148v^12 - 378480v^10 + 1425447v^8 - 4067664v^6 + 6150300v^4 - 8357500*v^2 + 4996625) / 3170625
$\beta_{6}$	$=$	$ ( 831 \nu^{14} - 5647 \nu^{12} + 31350 \nu^{10} + 79902 \nu^{8} - 618129 \nu^{6} + 2318760 \nu^{4} + \cdots + 2262875 ) / 634125 $ (831v^14 - 5647v^12 + 31350v^10 + 79902v^8 - 618129v^6 + 2318760v^4 - 2891475*v^2 + 2262875) / 634125
$\beta_{7}$	$=$	$ ( 6679 \nu^{15} - 104788 \nu^{13} + 934230 \nu^{11} - 4799832 \nu^{9} + 14815809 \nu^{7} + \cdots - 16577875 \nu ) / 3170625 $ (6679v^15 - 104788v^13 + 934230v^11 - 4799832v^9 + 14815809v^7 - 29322225v^5 + 32657500v^3 - 16577875v) / 3170625
$\beta_{8}$	$=$	$ ( -12\nu^{14} + 119\nu^{12} - 950\nu^{10} + 2996\nu^{8} - 8842\nu^{6} + 12730\nu^{4} - 19550\nu^{2} + 7750 ) / 7125 $ (-12v^14 + 119v^12 - 950v^10 + 2996v^8 - 8842v^6 + 12730v^4 - 19550*v^2 + 7750) / 7125
$\beta_{9}$	$=$	$ ( 8461 \nu^{15} - 107342 \nu^{13} + 894045 \nu^{11} - 3816413 \nu^{9} + 10830156 \nu^{7} + \cdots - 17051500 \nu ) / 3170625 $ (8461v^15 - 107342v^13 + 894045v^11 - 3816413v^9 + 10830156v^7 - 19654075v^5 + 23532250v^3 - 17051500v) / 3170625
$\beta_{10}$	$=$	$ ( - 2447 \nu^{15} + 25839 \nu^{13} - 206625 \nu^{11} + 696751 \nu^{9} - 1875177 \nu^{7} + \cdots - 983375 \nu ) / 634125 $ (-2447v^15 + 25839v^13 - 206625v^11 + 696751v^9 - 1875177v^7 + 2277530v^5 - 2220125v^3 - 983375v) / 634125
$\beta_{11}$	$=$	$ ( - 10471 \nu^{14} + 144287 \nu^{12} - 1218945 \nu^{10} + 5567493 \nu^{8} - 15642291 \nu^{6} + \cdots + 14057750 ) / 3170625 $ (-10471v^14 + 144287v^12 - 1218945v^10 + 5567493v^8 - 15642291v^6 + 28273425v^4 - 27410125*v^2 + 14057750) / 3170625
$\beta_{12}$	$=$	$ ( - 15193 \nu^{15} + 176571 \nu^{13} - 1460910 \nu^{11} + 5690969 \nu^{9} - 16206903 \nu^{7} + \cdots + 4211375 \nu ) / 3170625 $ (-15193v^15 + 176571v^13 - 1460910v^11 + 5690969v^9 - 16206903v^7 + 24014575v^5 - 26190625v^3 + 4211375v) / 3170625
$\beta_{13}$	$=$	$ ( 12874 \nu^{14} - 123728 \nu^{12} + 939930 \nu^{10} - 2489517 \nu^{8} + 5326479 \nu^{6} + \cdots + 7535875 ) / 3170625 $ (12874v^14 - 123728v^12 + 939930v^10 - 2489517v^8 + 5326479v^6 - 1132875v^4 - 2093375*v^2 + 7535875) / 3170625
$\beta_{14}$	$=$	$ ( 14343 \nu^{14} - 183596 \nu^{12} + 1544985 \nu^{10} - 6645669 \nu^{8} + 18837978 \nu^{6} + \cdots - 11347625 ) / 3170625 $ (14343v^14 - 183596v^12 + 1544985v^10 - 6645669v^8 + 18837978v^6 - 30781425v^4 + 28939125*v^2 - 11347625) / 3170625
$\beta_{15}$	$=$	$ ( 19846 \nu^{15} - 243562 \nu^{13} + 2011245 \nu^{11} - 8254868 \nu^{9} + 22837116 \nu^{7} + \cdots - 19270875 \nu ) / 3170625 $ (19846v^15 - 243562v^13 + 2011245v^11 - 8254868v^9 + 22837116v^7 - 37808575v^5 + 37381375v^3 - 19270875v) / 3170625

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

$\nu$	$=$	$ ( \beta_{15} + \beta_{10} - \beta_{9} + \beta_{7} + \beta_{4} - 2\beta_{3} - \beta_{2} ) / 3 $ (b15 + b10 - b9 + b7 + b4 - 2*b3 - b2) / 3
$\nu^{2}$	$=$	$ ( -\beta_{14} + \beta_{13} + \beta_{11} + 3\beta_{8} - 2\beta_{6} - 9\beta_{5} - 3\beta _1 + 9 ) / 3 $ (-b14 + b13 + b11 + 3b8 - 2b6 - 9b5 - 3b1 + 9) / 3
$\nu^{3}$	$=$	$ ( -4\beta_{15} + 3\beta_{12} - 4\beta_{10} + \beta_{9} + 11\beta_{7} - 7\beta_{4} - 4\beta_{3} + 7\beta_{2} ) / 3 $ (-4b15 + 3b12 - 4b10 + b9 + 11b7 - 7b4 - 4b3 + 7*b2) / 3
$\nu^{4}$	$=$	$ -6\beta_{14} + 6\beta_{13} - 4\beta_{11} + 14\beta_{8} - 4\beta_{6} - 14\beta_{5} + 7\beta_1 $ -6b14 + 6b13 - 4b11 + 14b8 - 4b6 - 14b5 + 7*b1
$\nu^{5}$	$=$	$ ( -67\beta_{15} + 51\beta_{12} - 97\beta_{10} + 46\beta_{9} + 47\beta_{7} - 67\beta_{4} + 47\beta_{3} + 46\beta_{2} ) / 3 $ (-67b15 + 51b12 - 97b10 + 46b9 + 47b7 - 67b4 + 47b3 + 46b2) / 3
$\nu^{6}$	$=$	$ ( -95\beta_{14} + 38\beta_{13} - 133\beta_{11} + 135\beta_{8} + 38\beta_{6} + 270\beta _1 - 243 ) / 3 $ (-95b14 + 38b13 - 133b11 + 135b8 + 38b6 + 270b1 - 243) / 3
$\nu^{7}$	$=$	$ ( - 311 \beta_{15} + 228 \beta_{12} - 482 \beta_{10} + 347 \beta_{9} - 119 \beta_{7} + \cdots - 52 \beta_{2} ) / 3 $ (-311b15 + 228b12 - 482b10 + 347b9 - 119b7 - 119b4 + 430b3 - 52b2) / 3
$\nu^{8}$	$=$	$ 84\beta_{14} - 224\beta_{13} - 84\beta_{11} - 292\beta_{8} + 308\beta_{6} + 511\beta_{5} + 292\beta _1 - 511 $ 84b14 - 224b13 - 84b11 - 292b8 + 308b6 + 511b5 + 292*b1 - 511
$\nu^{9}$	$=$	$ ( 763 \beta_{15} - 876 \beta_{12} + 1183 \beta_{10} + 413 \beta_{9} - 2822 \beta_{7} + \cdots - 2359 \beta_{2} ) / 3 $ (763b15 - 876b12 + 1183b10 + 413b9 - 2822b7 + 2059b4 + 763b3 - 2359b2) / 3
$\nu^{10}$	$=$	$ ( 6266\beta_{14} - 6266\beta_{13} + 4579\beta_{11} - 11514\beta_{8} + 4579\beta_{6} + 9999\beta_{5} - 5757\beta_1 ) / 3 $ (6266b14 - 6266b13 + 4579b11 - 11514b8 + 4579b6 + 9999b5 - 5757*b1) / 3
$\nu^{11}$	$=$	$ ( 18689 \beta_{15} - 16602 \beta_{12} + 29534 \beta_{10} - 12932 \beta_{9} - 13669 \beta_{7} + \cdots - 12932 \beta_{2} ) / 3 $ (18689b15 - 16602b12 + 29534b10 - 12932b9 - 13669b7 + 18689b4 - 13669b3 - 12932b2) / 3
$\nu^{12}$	$=$	$ 10260\beta_{14} - 3762\beta_{13} + 14022\beta_{11} - 12705\beta_{8} - 3762\beta_{6} - 25410\beta _1 + 22022 $ 10260b14 - 3762b13 + 14022b11 - 12705b8 - 3762b6 - 25410b1 + 22022
$\nu^{13}$	$=$	$ ( 90917 \beta_{15} - 72846 \beta_{12} + 144269 \beta_{10} - 106154 \beta_{9} + 33308 \beta_{7} + \cdots + 20044 \beta_{2} ) / 3 $ (90917b15 - 72846b12 + 144269b10 - 106154b9 + 33308b7 + 33308b4 - 124225b3 + 20044b2) / 3
$\nu^{14}$	$=$	$ ( - 75374 \beta_{14} + 205829 \beta_{13} + 75374 \beta_{11} + 253257 \beta_{8} - 281203 \beta_{6} + \cdots + 438741 ) / 3 $ (-75374b14 + 205829b13 + 75374b11 + 253257b8 - 281203b6 - 438741b5 - 253257*b1 + 438741) / 3
$\nu^{15}$	$=$	$ ( - 221621 \beta_{15} + 253257 \beta_{12} - 352076 \beta_{10} - 134956 \beta_{9} + \cdots + 708653 \beta_{2} ) / 3 $ (-221621b15 + 253257b12 - 352076b10 - 134956b9 + 826954b7 - 605333b4 - 221621b3 + 708653b2) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$\beta_{8}$	$\beta_{5}$

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} $

53.1

0.750156	+	0.433103i
1.15723	+	0.668129i
−1.15723	−	0.668129i
−0.750156	−	0.433103i
0.750156	−	0.433103i
1.15723	−	0.668129i
−1.15723	+	0.668129i
−0.750156	+	0.433103i
−1.44919	+	0.836690i
2.23560	−	1.29073i
−2.23560	+	1.29073i
1.44919	−	0.836690i
−1.44919	−	0.836690i
2.23560	+	1.29073i
−2.23560	−	1.29073i
1.44919	+	0.836690i

−0.657293

2.45305i

−3.85337

−

2.22474i

−1.44919

1.70289i

−3.03906

−

0.814313i

4.39869

−

4.39869i

−3.22474

−

4.67423i

53.2

−0.322280

1.20277i

0.389270

0.224745i

2.23560

0.0455319i

0.307007

0.0822623i

−2.15674

2.15674i

−0.775255

2.67423i

53.3

0.322280

−

1.20277i

0.389270

0.224745i

−2.23560

−

0.0455319i

0.307007

0.0822623i

2.15674

−

2.15674i

−0.775255

2.67423i

53.4

0.657293

−

2.45305i

−3.85337

−

2.22474i

1.44919

−

1.70289i

−3.03906

−

0.814313i

−4.39869

4.39869i

−3.22474

−

4.67423i

107.1

−0.657293

−

2.45305i

−3.85337

2.22474i

−1.44919

−

1.70289i

−3.03906

0.814313i

4.39869

4.39869i

−3.22474

4.67423i

107.2

−0.322280

−

1.20277i

0.389270

−

0.224745i

2.23560

−

0.0455319i

0.307007

−

0.0822623i

−2.15674

−

2.15674i

−0.775255

−

2.67423i

107.3

0.322280

1.20277i

0.389270

−

0.224745i

−2.23560

0.0455319i

0.307007

−

0.0822623i

2.15674

2.15674i

−0.775255

−

2.67423i

107.4

0.657293

2.45305i

−3.85337

2.22474i

1.44919

1.70289i

−3.03906

0.814313i

−4.39869

−

4.39869i

−3.22474

4.67423i

188.1

−2.45305

0.657293i

3.85337

−

2.22474i

0.750156

2.10648i

0.814313

3.03906i

−4.39869

4.39869i

−3.22474

−

4.67423i

188.2

−1.20277

0.322280i

−0.389270

0.224745i

1.15723

−

1.91332i

−0.0822623

−

0.307007i

2.15674

−

2.15674i

−0.775255

2.67423i

188.3

1.20277

−

0.322280i

−0.389270

0.224745i

−1.15723

1.91332i

−0.0822623

−

0.307007i

−2.15674

2.15674i

−0.775255

2.67423i

188.4

2.45305

−

0.657293i

3.85337

−

2.22474i

−0.750156

−

2.10648i

0.814313

3.03906i

4.39869

−

4.39869i

−3.22474

−

4.67423i

377.1

−2.45305

−

0.657293i

3.85337

2.22474i

0.750156

−

2.10648i

0.814313

−

3.03906i

−4.39869

−

4.39869i

−3.22474

4.67423i

377.2

−1.20277

−

0.322280i

−0.389270

−

0.224745i

1.15723

1.91332i

−0.0822623

0.307007i

2.15674

2.15674i

−0.775255

−

2.67423i

377.3

1.20277

0.322280i

−0.389270

−

0.224745i

−1.15723

−

1.91332i

−0.0822623

0.307007i

−2.15674

−

2.15674i

−0.775255

−

2.67423i

377.4

2.45305

0.657293i

3.85337

2.22474i

−0.750156

2.10648i

0.814313

−

3.03906i

4.39869

4.39869i

−3.22474

4.67423i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
15.e	even	4	1	inner
45.k	odd	12	1	inner
45.l	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	405.2.m.b		16
3.b	odd	2	1	inner	405.2.m.b		16
5.c	odd	4	1	inner	405.2.m.b		16
9.c	even	3	1		135.2.f.b	✓	8
9.c	even	3	1	inner	405.2.m.b		16
9.d	odd	6	1		135.2.f.b	✓	8
9.d	odd	6	1	inner	405.2.m.b		16
15.e	even	4	1	inner	405.2.m.b		16
36.f	odd	6	1		2160.2.w.a		8
36.h	even	6	1		2160.2.w.a		8
45.h	odd	6	1		675.2.f.g		8
45.j	even	6	1		675.2.f.g		8
45.k	odd	12	1		135.2.f.b	✓	8
45.k	odd	12	1	inner	405.2.m.b		16
45.k	odd	12	1		675.2.f.g		8
45.l	even	12	1		135.2.f.b	✓	8
45.l	even	12	1	inner	405.2.m.b		16
45.l	even	12	1		675.2.f.g		8
180.v	odd	12	1		2160.2.w.a		8
180.x	even	12	1		2160.2.w.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.2.f.b	✓	8	9.c	even	3	1
135.2.f.b	✓	8	9.d	odd	6	1
135.2.f.b	✓	8	45.k	odd	12	1
135.2.f.b	✓	8	45.l	even	12	1
405.2.m.b		16	1.a	even	1	1	trivial
405.2.m.b		16	3.b	odd	2	1	inner
405.2.m.b		16	5.c	odd	4	1	inner
405.2.m.b		16	9.c	even	3	1	inner
405.2.m.b		16	9.d	odd	6	1	inner
405.2.m.b		16	15.e	even	4	1	inner
405.2.m.b		16	45.k	odd	12	1	inner
405.2.m.b		16	45.l	even	12	1	inner
675.2.f.g		8	45.h	odd	6	1
675.2.f.g		8	45.j	even	6	1
675.2.f.g		8	45.k	odd	12	1
675.2.f.g		8	45.l	even	12	1
2160.2.w.a		8	36.f	odd	6	1
2160.2.w.a		8	36.h	even	6	1
2160.2.w.a		8	180.v	odd	12	1
2160.2.w.a		8	180.x	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{16} - 44T_{2}^{12} + 1836T_{2}^{8} - 4400T_{2}^{4} + 10000 $ T2^16 - 44*T2^12 + 1836*T2^8 - 4400*T2^4 + 10000 acting on $S_{2}^{\mathrm{new}}(405, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - 44 T^{12} + \cdots + 10000 $ T^16 - 44T^12 + 1836T^8 - 4400*T^4 + 10000
$3$	$ T^{16} $ T^16
$5$	$ T^{16} + 4 T^{14} + \cdots + 390625 $ T^16 + 4T^14 + 16T^12 - 200T^10 - 1025T^8 - 5000T^6 + 10000T^4 + 62500*T^2 + 390625
$7$	$ (T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 1)^{2} $ (T^8 + 4T^7 + 8T^6 + 40T^5 + 79T^4 - 40T^3 + 8T^2 - 4*T + 1)^2
$11$	$ (T^{8} - 16 T^{6} + \cdots + 100)^{2} $ (T^8 - 16T^6 + 246T^4 - 160*T^2 + 100)^2
$13$	$ (T^{8} - 8 T^{7} + \cdots + 625)^{2} $ (T^8 - 8T^7 + 32T^6 - 176T^5 + 679T^4 - 880T^3 + 800T^2 - 1000*T + 625)^2
$17$	$ (T^{8} + 944 T^{4} + 1600)^{2} $ (T^8 + 944*T^4 + 1600)^2
$19$	$ (T^{2} + 9)^{8} $ (T^2 + 9)^8
$23$	$ T^{16} - 44 T^{12} + \cdots + 10000 $ T^16 - 44T^12 + 1836T^8 - 4400*T^4 + 10000
$29$	$ (T^{8} + 96 T^{6} + \cdots + 5062500)^{2} $ (T^8 + 96T^6 + 6966T^4 + 216000*T^2 + 5062500)^2
$31$	$ (T^{4} + 4 T^{3} + \cdots + 2500)^{4} $ (T^4 + 4T^3 + 66T^2 - 200*T + 2500)^4
$37$	$ (T^{4} + 8 T^{3} + \cdots + 361)^{4} $ (T^4 + 8T^3 + 32T^2 - 152*T + 361)^4
$41$	$ (T^{8} - 136 T^{6} + \cdots + 62500)^{2} $ (T^8 - 136T^6 + 18246T^4 - 34000*T^2 + 62500)^2
$43$	$ (T^{8} + 4 T^{7} + \cdots + 10000)^{2} $ (T^8 + 4T^7 + 8T^6 + 112T^5 + 124T^4 - 1120T^3 + 800T^2 - 4000*T + 10000)^2
$47$	$ T^{16} + \cdots + 42949672960000 $ T^16 - 11264T^12 + 120324096T^8 - 73819750400*T^4 + 42949672960000
$53$	$ (T^{8} + 12524 T^{4} + 27984100)^{2} $ (T^8 + 12524*T^4 + 27984100)^2
$59$	$ (T^{8} + 240 T^{6} + \cdots + 81000000)^{2} $ (T^8 + 240T^6 + 48600T^4 + 2160000*T^2 + 81000000)^2
$61$	$ (T^{2} - T + 1)^{8} $ (T^2 - T + 1)^8
$67$	$ (T^{8} - 20 T^{7} + \cdots + 4879681)^{2} $ (T^8 - 20T^7 + 200T^6 - 2120T^5 + 18991T^4 - 99640T^3 + 441800T^2 - 2076460*T + 4879681)^2
$71$	$ (T^{4} + 184 T^{2} + 8410)^{4} $ (T^4 + 184*T^2 + 8410)^4
$73$	$ (T^{4} - 4 T^{3} + \cdots + 625)^{4} $ (T^4 - 4T^3 + 8T^2 + 100*T + 625)^4
$79$	$ (T^{8} - 30 T^{6} + \cdots + 81)^{2} $ (T^8 - 30T^6 + 891T^4 - 270*T^2 + 81)^2
$83$	$ T^{16} + \cdots + 3906250000 $ T^16 - 524T^12 + 212076T^8 - 32750000*T^4 + 3906250000
$89$	$ (T^{4} - 144 T^{2} + 3240)^{4} $ (T^4 - 144*T^2 + 3240)^4
$97$	$ (T^{8} + 16 T^{7} + \cdots + 3418801)^{2} $ (T^8 + 16T^7 + 128T^6 + 3424T^5 + 25543T^4 - 147232T^3 + 236672T^2 - 1272112*T + 3418801)^2
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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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.m (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{10} + \beta_{9}) q^{2} + ( - \beta_{14} - \beta_{13} - 2 \beta_1) q^{4} + \beta_{7} q^{5} + ( - \beta_{14} - \beta_{8} - \beta_{6} + \cdots - 1) q^{7}+ \cdots + (\beta_{12} + \beta_{10} + \cdots - \beta_{2}) q^{8}+O(q^{10}) \) q + (-b10 + b9) * q^2 + (-b14 - b13 - 2b1) q^4 + b7 * q^5 + (-b14 - b8 - b6 + b5 - b1 - 1) * q^7 + (b12 + b10 - 2b3 - b2) q^8	\( q + ( - \beta_{10} + \beta_{9}) q^{2} + ( - \beta_{14} - \beta_{13} - 2 \beta_1) q^{4} + \beta_{7} q^{5} + ( - \beta_{14} - \beta_{8} - \beta_{6} + \cdots - 1) q^{7}+ \cdots + ( - 2 \beta_{10} + 4 \beta_{3}) q^{98}+O(q^{100}) \) q + (-b10 + b9) * q^2 + (-b14 - b13 - 2b1) q^4 + b7 * q^5 + (-b14 - b8 - b6 + b5 - b1 - 1) * q^7 + (b12 + b10 - 2b3 - b2) q^8 + (-2b13 + 2b11 + b8 + b6 - 2) * q^10 + (-b15 - b10 + b9 - b4 + b2) * q^11 + (-b13 + 2b5 + 2b1) * q^13 + (b12 + b10 - 2b9 - b7 + b4 - b3) q^14 + (2b14 - 2b11 + 2b6 - 2b5 + 2) * q^16 + (-2b15 + b12 - b10 + 2b7 + b2) * q^17 + 3b8 q^19 + (-b15 + b10 - b9 - b4 - 2b2) q^20 + (2b14 + b5 - b1) q^22 - b12 * q^23 + (-2b14 + b11 + 3b8 - 2b6 + b5 + 3b1 - 1) * q^25 + (-b15 - b12 - b10 + b7 - b3 + 2b2) q^26 + (4b13 - 4b11 - 5b8 + 5) q^28 + (b15 + 2b10 - 2b9 - b4 + 2b2) q^29 + (-3b14 + 3b13 - 2b5) q^31 + 2b9 q^32 + (2b14 + 2b11 - 2b8 + 2b6 - 2b1) q^34 + (2b12 - b3 - 2b2) * q^35 + (-2b8 - 3b6 - 2) * q^37 - 3b2 q^38 + (3b14 + b13 - 9b5 + 3b1) q^40 + (3b12 + b10 + 2b9 + b7 + b4 - b3) * q^41 + (2b11 + b8 + b5 + b1 - 1) q^43 + (-b12 + b10 + b2) * q^44 + (-b13 + b11 - b6 - 4) * q^46 + (-4b10 + 4b9) * q^47 + (2b14 + 2b13 - 2b1) q^49 + (-4b12 + b10 - 3b9 - 3b7 + b4 - b3) q^50 + (-2b14 - b8 - 2b6 + b5 - b1 - 1) * q^52 + (-b12 - 2b10 + 4b3 + b2) * q^53 + (b13 - b11 - 3b8 + 2b6 - 4) * q^55 + (2b15 - 3b10 + 3b9 + 2b4 + 3b2) q^56 + (2b13 + 9b5 + 9b1) q^58 + (3b12 - 4b10 + b9 + 4b7 - 4b4 + 4b3) q^59 + (-b5 + 1) * q^61 + (6b15 - 3b12 - b10 - 6b7 - 3b2) * q^62 + (2b13 - 2b11 + 4b8 - 2b6) * q^64 + (2b15 + 2b10 - 2b9 + 3b4 - 3b2) q^65 + (-b14 + 5b5 - 5b1) * q^67 - 2b12 q^68 + (3b14 - 4b11 - 2b8 + 3b6 - 9b5 - 2b1 + 9) * q^70 + (b15 + 3b12 + 3b10 - b7 + b3 - 4b2) q^71 + (-3b13 + 3b11 - b8 + 1) * q^73 + (-3b15 + 5b10 - 5b9 + 3b4 + 5b2) q^74 + (3b14 - 3b13 - 6b5) q^76 + b9 * q^77 + (-b14 - b11 + 3b8 - b6 + 3b1) * q^79 + (-2b12 + 4b10 + 2b3 + 2b2) * q^80 + (11b8 + 8b6 + 11) * q^82 + (-b10 + b9 - 2b4) q^83 + (-3b14 - b13 - b5 + 7b1) * q^85 + (-3b12 - 2b10 - b9 - 2b7 - 2b4 + 2b3) q^86 + (-2b11 + 6b8 + 6b5 + 6b1 - 6) * q^88 + (-3b12 + 3b10 + 3b2) q^89 + (-b13 + b11 - b6 - 1) * q^91 + (-2b15 + 3b10 - 3b9 + b2) q^92 + (-4b14 - 4b13 - 16b1) q^94 + (-3b10 - 3b4 + 3b3) q^95 + (5b14 - 4b8 + 5b6 + 4b5 - 4b1 - 4) q^97 + (-2b10 + 4b3) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{7}+O(q^{10}) \) 16 * q - 8 * q^7	\( 16 q - 8 q^{7} - 32 q^{10} + 16 q^{13} + 16 q^{16} + 8 q^{22} - 8 q^{25} + 80 q^{28} - 16 q^{31} - 32 q^{37} - 72 q^{40} - 8 q^{43} - 64 q^{46} - 8 q^{52} - 64 q^{55} + 72 q^{58} + 8 q^{61} + 40 q^{67} + 72 q^{70} + 16 q^{73} - 48 q^{76} + 176 q^{82} - 8 q^{85} - 48 q^{88} - 16 q^{91} - 32 q^{97}+O(q^{100}) \) 16 * q - 8 * q^7 - 32 * q^10 + 16 * q^13 + 16 * q^16 + 8 * q^22 - 8 * q^25 + 80 * q^28 - 16 * q^31 - 32 * q^37 - 72 * q^40 - 8 * q^43 - 64 * q^46 - 8 * q^52 - 64 * q^55 + 72 * q^58 + 8 * q^61 + 40 * q^67 + 72 * q^70 + 16 * q^73 - 48 * q^76 + 176 * q^82 - 8 * q^85 - 48 * q^88 - 16 * q^91 - 32 * q^97

Introduction

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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	405.2.m.b

Level	$405$
Weight	$2$
Character orbit	405.m
Analytic conductor	$3.234$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(3.23394128186\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 12x^{14} + 100x^{12} - 408x^{10} + 1191x^{8} - 2040x^{6} + 2500x^{4} - 1500x^{2} + 625 \) x^16 - 12x^14 + 100x^12 - 408x^10 + 1191x^8 - 2040x^6 + 2500x^4 - 1500*x^2 + 625
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{4} \)
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(\beta_{1}\)	\(=\)	\( ( 162 \nu^{14} + 1386 \nu^{12} + 190 \nu^{10} + 31754 \nu^{8} + 610802 \nu^{6} - 1207925 \nu^{4} + \cdots + 1408250 ) / 3170625 \) (162v^14 + 1386v^12 + 190v^10 + 31754v^8 + 610802v^6 - 1207925v^4 + 3359500*v^2 + 1408250) / 3170625
\(\beta_{2}\)	\(=\)	\( ( - 2373 \nu^{15} + 25681 \nu^{13} - 196935 \nu^{11} + 583909 \nu^{9} - 895008 \nu^{7} + \cdots - 11788000 \nu ) / 3170625 \) (-2373v^15 + 25681v^13 - 196935v^11 + 583909v^9 - 895008v^7 - 1925650v^5 + 5332125v^3 - 11788000v) / 3170625
\(\beta_{3}\)	\(=\)	\( ( 519 \nu^{15} - 11728 \nu^{13} + 111150 \nu^{11} - 674877 \nu^{9} + 2149629 \nu^{7} + \cdots - 3584500 \nu ) / 634125 \) (519v^15 - 11728v^13 + 111150v^11 - 674877v^9 + 2149629v^7 - 4634385v^5 + 4860000v^3 - 3584500v) / 634125
\(\beta_{4}\)	\(=\)	\( ( - 3012 \nu^{15} + 20214 \nu^{13} - 103740 \nu^{11} - 410329 \nu^{9} + 3154398 \nu^{7} + \cdots - 14407000 \nu ) / 3170625 \) (-3012v^15 + 20214v^13 - 103740v^11 - 410329v^9 + 3154398v^7 - 12180425v^5 + 18526125v^3 - 14407000v) / 3170625
\(\beta_{5}\)	\(=\)	\( ( - 4084 \nu^{14} + 46148 \nu^{12} - 378480 \nu^{10} + 1425447 \nu^{8} - 4067664 \nu^{6} + \cdots + 4996625 ) / 3170625 \) (-4084v^14 + 46148v^12 - 378480v^10 + 1425447v^8 - 4067664v^6 + 6150300v^4 - 8357500*v^2 + 4996625) / 3170625
\(\beta_{6}\)	\(=\)	\( ( 831 \nu^{14} - 5647 \nu^{12} + 31350 \nu^{10} + 79902 \nu^{8} - 618129 \nu^{6} + 2318760 \nu^{4} + \cdots + 2262875 ) / 634125 \) (831v^14 - 5647v^12 + 31350v^10 + 79902v^8 - 618129v^6 + 2318760v^4 - 2891475*v^2 + 2262875) / 634125
\(\beta_{7}\)	\(=\)	\( ( 6679 \nu^{15} - 104788 \nu^{13} + 934230 \nu^{11} - 4799832 \nu^{9} + 14815809 \nu^{7} + \cdots - 16577875 \nu ) / 3170625 \) (6679v^15 - 104788v^13 + 934230v^11 - 4799832v^9 + 14815809v^7 - 29322225v^5 + 32657500v^3 - 16577875v) / 3170625
\(\beta_{8}\)	\(=\)	\( ( -12\nu^{14} + 119\nu^{12} - 950\nu^{10} + 2996\nu^{8} - 8842\nu^{6} + 12730\nu^{4} - 19550\nu^{2} + 7750 ) / 7125 \) (-12v^14 + 119v^12 - 950v^10 + 2996v^8 - 8842v^6 + 12730v^4 - 19550*v^2 + 7750) / 7125
\(\beta_{9}\)	\(=\)	\( ( 8461 \nu^{15} - 107342 \nu^{13} + 894045 \nu^{11} - 3816413 \nu^{9} + 10830156 \nu^{7} + \cdots - 17051500 \nu ) / 3170625 \) (8461v^15 - 107342v^13 + 894045v^11 - 3816413v^9 + 10830156v^7 - 19654075v^5 + 23532250v^3 - 17051500v) / 3170625
\(\beta_{10}\)	\(=\)	\( ( - 2447 \nu^{15} + 25839 \nu^{13} - 206625 \nu^{11} + 696751 \nu^{9} - 1875177 \nu^{7} + \cdots - 983375 \nu ) / 634125 \) (-2447v^15 + 25839v^13 - 206625v^11 + 696751v^9 - 1875177v^7 + 2277530v^5 - 2220125v^3 - 983375v) / 634125
\(\beta_{11}\)	\(=\)	\( ( - 10471 \nu^{14} + 144287 \nu^{12} - 1218945 \nu^{10} + 5567493 \nu^{8} - 15642291 \nu^{6} + \cdots + 14057750 ) / 3170625 \) (-10471v^14 + 144287v^12 - 1218945v^10 + 5567493v^8 - 15642291v^6 + 28273425v^4 - 27410125*v^2 + 14057750) / 3170625
\(\beta_{12}\)	\(=\)	\( ( - 15193 \nu^{15} + 176571 \nu^{13} - 1460910 \nu^{11} + 5690969 \nu^{9} - 16206903 \nu^{7} + \cdots + 4211375 \nu ) / 3170625 \) (-15193v^15 + 176571v^13 - 1460910v^11 + 5690969v^9 - 16206903v^7 + 24014575v^5 - 26190625v^3 + 4211375v) / 3170625
\(\beta_{13}\)	\(=\)	\( ( 12874 \nu^{14} - 123728 \nu^{12} + 939930 \nu^{10} - 2489517 \nu^{8} + 5326479 \nu^{6} + \cdots + 7535875 ) / 3170625 \) (12874v^14 - 123728v^12 + 939930v^10 - 2489517v^8 + 5326479v^6 - 1132875v^4 - 2093375*v^2 + 7535875) / 3170625
\(\beta_{14}\)	\(=\)	\( ( 14343 \nu^{14} - 183596 \nu^{12} + 1544985 \nu^{10} - 6645669 \nu^{8} + 18837978 \nu^{6} + \cdots - 11347625 ) / 3170625 \) (14343v^14 - 183596v^12 + 1544985v^10 - 6645669v^8 + 18837978v^6 - 30781425v^4 + 28939125*v^2 - 11347625) / 3170625
\(\beta_{15}\)	\(=\)	\( ( 19846 \nu^{15} - 243562 \nu^{13} + 2011245 \nu^{11} - 8254868 \nu^{9} + 22837116 \nu^{7} + \cdots - 19270875 \nu ) / 3170625 \) (19846v^15 - 243562v^13 + 2011245v^11 - 8254868v^9 + 22837116v^7 - 37808575v^5 + 37381375v^3 - 19270875v) / 3170625

\(\nu\)	\(=\)	\( ( \beta_{15} + \beta_{10} - \beta_{9} + \beta_{7} + \beta_{4} - 2\beta_{3} - \beta_{2} ) / 3 \) (b15 + b10 - b9 + b7 + b4 - 2*b3 - b2) / 3
\(\nu^{2}\)	\(=\)	\( ( -\beta_{14} + \beta_{13} + \beta_{11} + 3\beta_{8} - 2\beta_{6} - 9\beta_{5} - 3\beta _1 + 9 ) / 3 \) (-b14 + b13 + b11 + 3b8 - 2b6 - 9b5 - 3b1 + 9) / 3
\(\nu^{3}\)	\(=\)	\( ( -4\beta_{15} + 3\beta_{12} - 4\beta_{10} + \beta_{9} + 11\beta_{7} - 7\beta_{4} - 4\beta_{3} + 7\beta_{2} ) / 3 \) (-4b15 + 3b12 - 4b10 + b9 + 11b7 - 7b4 - 4b3 + 7*b2) / 3
\(\nu^{4}\)	\(=\)	\( -6\beta_{14} + 6\beta_{13} - 4\beta_{11} + 14\beta_{8} - 4\beta_{6} - 14\beta_{5} + 7\beta_1 \) -6b14 + 6b13 - 4b11 + 14b8 - 4b6 - 14b5 + 7*b1
\(\nu^{5}\)	\(=\)	\( ( -67\beta_{15} + 51\beta_{12} - 97\beta_{10} + 46\beta_{9} + 47\beta_{7} - 67\beta_{4} + 47\beta_{3} + 46\beta_{2} ) / 3 \) (-67b15 + 51b12 - 97b10 + 46b9 + 47b7 - 67b4 + 47b3 + 46b2) / 3
\(\nu^{6}\)	\(=\)	\( ( -95\beta_{14} + 38\beta_{13} - 133\beta_{11} + 135\beta_{8} + 38\beta_{6} + 270\beta _1 - 243 ) / 3 \) (-95b14 + 38b13 - 133b11 + 135b8 + 38b6 + 270b1 - 243) / 3
\(\nu^{7}\)	\(=\)	\( ( - 311 \beta_{15} + 228 \beta_{12} - 482 \beta_{10} + 347 \beta_{9} - 119 \beta_{7} + \cdots - 52 \beta_{2} ) / 3 \) (-311b15 + 228b12 - 482b10 + 347b9 - 119b7 - 119b4 + 430b3 - 52b2) / 3
\(\nu^{8}\)	\(=\)	\( 84\beta_{14} - 224\beta_{13} - 84\beta_{11} - 292\beta_{8} + 308\beta_{6} + 511\beta_{5} + 292\beta _1 - 511 \) 84b14 - 224b13 - 84b11 - 292b8 + 308b6 + 511b5 + 292*b1 - 511
\(\nu^{9}\)	\(=\)	\( ( 763 \beta_{15} - 876 \beta_{12} + 1183 \beta_{10} + 413 \beta_{9} - 2822 \beta_{7} + \cdots - 2359 \beta_{2} ) / 3 \) (763b15 - 876b12 + 1183b10 + 413b9 - 2822b7 + 2059b4 + 763b3 - 2359b2) / 3
\(\nu^{10}\)	\(=\)	\( ( 6266\beta_{14} - 6266\beta_{13} + 4579\beta_{11} - 11514\beta_{8} + 4579\beta_{6} + 9999\beta_{5} - 5757\beta_1 ) / 3 \) (6266b14 - 6266b13 + 4579b11 - 11514b8 + 4579b6 + 9999b5 - 5757*b1) / 3
\(\nu^{11}\)	\(=\)	\( ( 18689 \beta_{15} - 16602 \beta_{12} + 29534 \beta_{10} - 12932 \beta_{9} - 13669 \beta_{7} + \cdots - 12932 \beta_{2} ) / 3 \) (18689b15 - 16602b12 + 29534b10 - 12932b9 - 13669b7 + 18689b4 - 13669b3 - 12932b2) / 3
\(\nu^{12}\)	\(=\)	\( 10260\beta_{14} - 3762\beta_{13} + 14022\beta_{11} - 12705\beta_{8} - 3762\beta_{6} - 25410\beta _1 + 22022 \) 10260b14 - 3762b13 + 14022b11 - 12705b8 - 3762b6 - 25410b1 + 22022
\(\nu^{13}\)	\(=\)	\( ( 90917 \beta_{15} - 72846 \beta_{12} + 144269 \beta_{10} - 106154 \beta_{9} + 33308 \beta_{7} + \cdots + 20044 \beta_{2} ) / 3 \) (90917b15 - 72846b12 + 144269b10 - 106154b9 + 33308b7 + 33308b4 - 124225b3 + 20044b2) / 3
\(\nu^{14}\)	\(=\)	\( ( - 75374 \beta_{14} + 205829 \beta_{13} + 75374 \beta_{11} + 253257 \beta_{8} - 281203 \beta_{6} + \cdots + 438741 ) / 3 \) (-75374b14 + 205829b13 + 75374b11 + 253257b8 - 281203b6 - 438741b5 - 253257*b1 + 438741) / 3
\(\nu^{15}\)	\(=\)	\( ( - 221621 \beta_{15} + 253257 \beta_{12} - 352076 \beta_{10} - 134956 \beta_{9} + \cdots + 708653 \beta_{2} ) / 3 \) (-221621b15 + 253257b12 - 352076b10 - 134956b9 + 826954b7 - 605333b4 - 221621b3 + 708653b2) / 3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 53.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - 44 T^{12} + \cdots + 10000 \) T^16 - 44T^12 + 1836T^8 - 4400*T^4 + 10000
$3$	\( T^{16} \) T^16
$5$	\( T^{16} + 4 T^{14} + \cdots + 390625 \) T^16 + 4T^14 + 16T^12 - 200T^10 - 1025T^8 - 5000T^6 + 10000T^4 + 62500*T^2 + 390625
$7$	\( (T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 1)^{2} \) (T^8 + 4T^7 + 8T^6 + 40T^5 + 79T^4 - 40T^3 + 8T^2 - 4*T + 1)^2
$11$	\( (T^{8} - 16 T^{6} + \cdots + 100)^{2} \) (T^8 - 16T^6 + 246T^4 - 160*T^2 + 100)^2
$13$	\( (T^{8} - 8 T^{7} + \cdots + 625)^{2} \) (T^8 - 8T^7 + 32T^6 - 176T^5 + 679T^4 - 880T^3 + 800T^2 - 1000*T + 625)^2
$17$	\( (T^{8} + 944 T^{4} + 1600)^{2} \) (T^8 + 944*T^4 + 1600)^2
$19$	\( (T^{2} + 9)^{8} \) (T^2 + 9)^8
$23$	\( T^{16} - 44 T^{12} + \cdots + 10000 \) T^16 - 44T^12 + 1836T^8 - 4400*T^4 + 10000
$29$	\( (T^{8} + 96 T^{6} + \cdots + 5062500)^{2} \) (T^8 + 96T^6 + 6966T^4 + 216000*T^2 + 5062500)^2
$31$	\( (T^{4} + 4 T^{3} + \cdots + 2500)^{4} \) (T^4 + 4T^3 + 66T^2 - 200*T + 2500)^4
$37$	\( (T^{4} + 8 T^{3} + \cdots + 361)^{4} \) (T^4 + 8T^3 + 32T^2 - 152*T + 361)^4
$41$	\( (T^{8} - 136 T^{6} + \cdots + 62500)^{2} \) (T^8 - 136T^6 + 18246T^4 - 34000*T^2 + 62500)^2
$43$	\( (T^{8} + 4 T^{7} + \cdots + 10000)^{2} \) (T^8 + 4T^7 + 8T^6 + 112T^5 + 124T^4 - 1120T^3 + 800T^2 - 4000*T + 10000)^2
$47$	\( T^{16} + \cdots + 42949672960000 \) T^16 - 11264T^12 + 120324096T^8 - 73819750400*T^4 + 42949672960000
$53$	\( (T^{8} + 12524 T^{4} + 27984100)^{2} \) (T^8 + 12524*T^4 + 27984100)^2
$59$	\( (T^{8} + 240 T^{6} + \cdots + 81000000)^{2} \) (T^8 + 240T^6 + 48600T^4 + 2160000*T^2 + 81000000)^2
$61$	\( (T^{2} - T + 1)^{8} \) (T^2 - T + 1)^8
$67$	\( (T^{8} - 20 T^{7} + \cdots + 4879681)^{2} \) (T^8 - 20T^7 + 200T^6 - 2120T^5 + 18991T^4 - 99640T^3 + 441800T^2 - 2076460*T + 4879681)^2
$71$	\( (T^{4} + 184 T^{2} + 8410)^{4} \) (T^4 + 184*T^2 + 8410)^4
$73$	\( (T^{4} - 4 T^{3} + \cdots + 625)^{4} \) (T^4 - 4T^3 + 8T^2 + 100*T + 625)^4
$79$	\( (T^{8} - 30 T^{6} + \cdots + 81)^{2} \) (T^8 - 30T^6 + 891T^4 - 270*T^2 + 81)^2
$83$	\( T^{16} + \cdots + 3906250000 \) T^16 - 524T^12 + 212076T^8 - 32750000*T^4 + 3906250000
$89$	\( (T^{4} - 144 T^{2} + 3240)^{4} \) (T^4 - 144*T^2 + 3240)^4
$97$	\( (T^{8} + 16 T^{7} + \cdots + 3418801)^{2} \) (T^8 + 16T^7 + 128T^6 + 3424T^5 + 25543T^4 - 147232T^3 + 236672T^2 - 1272112*T + 3418801)^2
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\(n\)	\(82\)	\(326\)
\(\chi(n)\)	\(\beta_{8}\)	\(\beta_{5}\)