LMFDB - Newform orbit 250.2.e.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [250,2,Mod(49,250)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(250, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([7]))

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

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

chi := DirichletCharacter("250.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 250 = 2 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	250.e (of order $10$, 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:	$1.99626005053$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
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} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 $ x^16 + x^14 - 4x^12 - 49x^10 + 11x^8 + 395x^6 + 900x^4 + 1125x^2 + 625
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	no (minimal twist has level 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-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_{12} + \cdots + \beta_{9}) q^{2} + ( - \beta_{14} + \beta_{13} - \beta_{9}) q^{3} - \beta_{3} q^{4} + (\beta_{8} + \beta_{6} + \cdots + \beta_{3}) q^{6} + ( - \beta_{14} - \beta_{11} + \cdots + \beta_{5}) q^{7}+ \cdots + ( - 2 \beta_{8} - \beta_{4} - \beta_{3} + \cdots + 6) q^{99}+O(q^{100}) $ q + (-b13 + b12 + b10 + b9) * q^2 + (-b14 + b13 - b9) * q^3 - b3 * q^4 + (b8 + b6 + b4 + b3) * q^6 + (-b14 - b11 - b9 + b5) * q^7 + b12 * q^8 + (-b8 - b7 - 3b6 - b4 - b2 - 1) q^9 + (b6 + b4 + b3 - b2 - b1 + 1) * q^11 + (-b15 + b13 - b12 - b9) * q^12 + (-b15 - b13 - 2b12 + b11 - 2b9 - b5) * q^13 + (b6 + b4 + b3 - b2 - b1 + 1) * q^14 + b6 * q^16 + (b15 - b14 + b13 - 3b12 - b10) q^17 + (b14 - b12 + b11 - b10 + 2b9) q^18 + (-b8 - b7 + 3b6 + 2b4 + 3b3 - b1) q^19 + (-5b6 - 4b3 + b1 - 5) * q^21 + (-b15 + b14 + b10 - b5) * q^22 + (2b15 - b14 - 3b13 + 3b12 - 2b11 + b10 + b9 + b5) * q^23 + (b4 + b3 - b2) * q^24 + (b8 + 2b4 + 2b3 + b1 + 3) * q^26 + (b14 - 5b13 + 5b12 + b10 + b9 - b5) * q^27 + (-b15 + b14 + b10 - b5) * q^28 + (2b6 - 3b3 - b1 + 2) * q^29 + (-2b8 - b7 + 2b6 - 3b4 + 2b3 - b1) * q^31 - b9 * q^32 + (5b13 - b12 + b11 - 5b10) * q^33 + (b8 - 3b6 - b4 + b2 - 1) q^34 + (-b8 - b7 - 3b6 - 3b4 - 2b3 - 2) q^36 + (2b15 + 2b13 + b12 + b9) * q^37 + (b15 + 2b13 - 3b12 - b11 - 3b9 + b5) q^38 + (b8 + b7 - 5b6 - 5b4 - 2b3 + 3b2 + 3b1 - 2) q^39 + (-b8 + b7 + 3b6 - b2) q^41 + (5b13 - b12 + b11 - 5b10) * q^42 + (-b14 + 4b12 - b11 + 4b10 + 3b9 + b5) q^43 + (b7 + b4 + b1) * q^44 + (b7 + 2b6 - b3 + b2 - b1 + 2) q^46 + (b15 - b14 - b10 + b5) * q^47 + (-b15 + b14 + b13 - b12 + b11 - b5) * q^48 + (b8 - 4b4 - 4b3 + b2 + b1 - 2) * q^49 + (-b8 + b4 + b3 + 2b2 - b1 + 5) q^51 + (-b15 - b13 + b12 + b11 + 3b10 + 3b9) * q^52 + (3b14 - 4b13 + 4b10 + 4b9) * q^53 + (b7 + 4b6 - b3 + b2 + b1 + 4) q^54 + (b7 + b4 + b1) * q^56 + (3b14 + 2b12 + 3b11 + 2b10 + 3b9 - b5) q^57 + (-2b13 + 5b12 - b11 + 2b10) q^58 + (2b8 + 4b7 + 2b6 + 2b2) * q^59 + (2b8 + 2b7 - 3b6 - 3b4 - 2b3 + 4b2 + 4b1 - 2) q^61 + (2b15 - 3b13 - 2b12 - b11 - 2b9 + b5) * q^62 + (b15 - 7b13 + b12 - 2b11 + b9 + 2b5) q^63 + (b6 + b4 + b3 + 1) * q^64 + (-b7 - b6 - 5b4 - 5) q^66 + (-3b15 + 3b14 - 7b12) q^67 + (-b14 - b12 - b11 - b10 + 2b9 + b5) q^68 + (3b8 + 2b7 + 7b6 + 13b4 + 7b3 + 2b1) * q^69 + (-b7 - 4b6 - 7b3 - b2 - b1 - 4) * q^71 + (b15 - b13 - 2b10 + b9 + b5) q^72 + (-3b15 + b14 + 8b13 - 8b12 + 3b11 - 6b10 - 6b9 - b5) * q^73 + (-b4 - b3 + 2b2 - 3) q^74 + (-b8 + 3b4 + 3b3 - b1 + 1) * q^76 + (-b14 + 5b13 - 5b12 - 9b10 - 9b9 + b5) * q^77 + (2b15 - 3b14 - 3b13 - 2b10 + 3b9 + 2b5) * q^78 + (2b6 + 4b3 + 2b1 + 2) q^79 + (2b8 + b7 + 6b6 + 6b3 + b1) q^81 + (b14 + b11 - 3b9 - 2b5) * q^82 + (-b15 + b14 + 2b13 - 7b12 - 2b11 - 2b10) * q^83 + (-b7 - b6 - 5b4 - 5) q^84 + (b6 + b4 - 3b3 - b2 - b1 - 3) q^86 + (-5b15 + 11b13 - 5b12 + 2b11 - 5b9 - 2b5) * q^87 + (b13 + b11 - b5) * q^88 + (-b8 - b7 - 2b6 - 2b4 + 2b3 - 4b2 - 4b1 + 2) q^89 + (3b8 - 2b7 - 3b6 - 2b4 + 3b2 - 2) q^91 + (b15 - b14 - 2b13 + 3b12 - 2b11 + 2b10) * q^92 + (2b14 + 7b12 + 2b11 + 7b10 - 2b9 - 5b5) * q^93 + (-b7 - b4 - b1) * q^94 + (-b6 + b1 - 1) * q^96 + (-5b13 - 3b10 + 5b9) q^97 + (-b14 - 2b13 + 2b12 - 2b10 - 2b9 + b5) * q^98 + (-2b8 - b4 - b3 - b2 - 2b1 + 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 2 q^{11} + 2 q^{14} - 4 q^{16} - 40 q^{19} - 38 q^{21} - 4 q^{24} + 44 q^{26} + 30 q^{29} - 18 q^{31} + 2 q^{34} - 2 q^{36} + 24 q^{39} - 18 q^{41} - 2 q^{44} + 14 q^{46}+ \cdots + 84 q^{99}+O(q^{100}) $ 16 * q + 4 * q^4 - 6 * q^6 + 2 * q^9 + 2 * q^11 + 2 * q^14 - 4 * q^16 - 40 * q^19 - 38 * q^21 - 4 * q^24 + 44 * q^26 + 30 * q^29 - 18 * q^31 + 2 * q^34 - 2 * q^36 + 24 * q^39 - 18 * q^41 - 2 * q^44 + 14 * q^46 + 8 * q^49 + 52 * q^51 + 50 * q^54 - 2 * q^56 - 20 * q^59 + 12 * q^61 + 4 * q^64 - 52 * q^66 - 86 * q^69 - 18 * q^71 - 48 * q^74 - 20 * q^76 + 20 * q^79 - 34 * q^81 - 52 * q^84 - 46 * q^86 + 30 * q^89 + 2 * q^91 + 2 * q^94 - 6 * q^96 + 84 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 $ x^16 + x^14 - 4*x^12 - 49*x^10 + 11*x^8 + 395*x^6 + 900*x^4 + 1125*x^2 + 625 :

$\beta_{1}$	$=$	$ ( 948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + \cdots + 181387875 ) / 171780125 $ (948392v^14 - 2081693v^12 - 1547103v^10 - 35207443v^8 + 136185777v^6 + 77053830v^4 + 27131400*v^2 + 181387875) / 171780125
$\beta_{2}$	$=$	$ ( 122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + \cdots + 52412250 ) / 15616375 $ (122987v^14 + 97172v^12 - 701513v^10 - 5799603v^8 + 3932417v^6 + 54634025v^4 + 77779875*v^2 + 52412250) / 15616375
$\beta_{3}$	$=$	$ ( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + \cdots + 563878625 ) / 171780125 $ (1379032v^14 - 312743v^12 - 4990978v^10 - 61900293v^8 + 94590252v^6 + 424939915v^4 + 771306350*v^2 + 563878625) / 171780125
$\beta_{4}$	$=$	$ ( - 281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} + \cdots - 167096475 ) / 34356025 $ (-281280v^14 + 69219v^12 + 939009v^10 + 12381889v^8 - 18275316v^6 - 83450271v^4 - 152170830*v^2 - 167096475) / 34356025
$\beta_{5}$	$=$	$ ( - 1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} + \cdots + 1705295750 \nu ) / 858900625 $ (-1157122v^15 + 1428203v^13 + 8678488v^11 + 43919978v^9 - 137248467v^7 - 474206065v^5 + 222668250v^3 + 1705295750v) / 858900625
$\beta_{6}$	$=$	$ ( - 441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} + \cdots - 215410900 ) / 34356025 $ (-441862v^14 + 108648v^12 + 1544473v^10 + 20140738v^8 - 29602957v^6 - 135351515v^4 - 246968035*v^2 - 215410900) / 34356025
$\beta_{7}$	$=$	$ ( - 2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} + \cdots - 860602375 ) / 171780125 $ (-2817591v^14 + 2379174v^12 + 8884104v^10 + 118381374v^8 - 262226761v^6 - 716617430v^4 - 884417625*v^2 - 860602375) / 171780125
$\beta_{8}$	$=$	$ ( - 626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} + \cdots - 222107450 ) / 34356025 $ (-626638v^14 + 144621v^12 + 2783831v^10 + 27217946v^8 - 41997039v^6 - 210462216v^4 - 281904075*v^2 - 222107450) / 34356025
$\beta_{9}$	$=$	$ ( 2293 \nu^{15} + 486 \nu^{13} - 12284 \nu^{11} - 101834 \nu^{9} + 117086 \nu^{7} + 906528 \nu^{5} + \cdots + 955700 \nu ) / 633875 $ (2293v^15 + 486v^13 - 12284v^11 - 101834v^9 + 117086v^7 + 906528v^5 + 1102430v^3 + 955700v) / 633875
$\beta_{10}$	$=$	$ ( 3183877 \nu^{15} - 4006568 \nu^{13} - 6543803 \nu^{11} - 134015793 \nu^{9} + \cdots + 1245786250 \nu ) / 858900625 $ (3183877v^15 - 4006568v^13 - 6543803v^11 - 134015793v^9 + 335534227v^7 + 577777045v^5 + 1177699300v^3 + 1245786250v) / 858900625
$\beta_{11}$	$=$	$ ( 8105189 \nu^{15} + 777904 \nu^{13} - 26062291 \nu^{11} - 376639746 \nu^{9} + \cdots + 5975622625 \nu ) / 858900625 $ (8105189v^15 + 777904v^13 - 26062291v^11 - 376639746v^9 + 389714419v^7 + 2537763645v^5 + 5645322900v^3 + 5975622625v) / 858900625
$\beta_{12}$	$=$	$ ( - 8189122 \nu^{15} + 3158678 \nu^{13} + 32153713 \nu^{11} + 353467203 \nu^{9} + \cdots - 3331016750 \nu ) / 858900625 $ (-8189122v^15 + 3158678v^13 + 32153713v^11 + 353467203v^9 - 594131367v^7 - 2560462840v^5 - 3581602500v^3 - 3331016750v) / 858900625
$\beta_{13}$	$=$	$ ( 8211971 \nu^{15} - 5580699 \nu^{13} - 25580004 \nu^{11} - 355005349 \nu^{9} + \cdots + 3295922875 \nu ) / 858900625 $ (8211971v^15 - 5580699v^13 - 25580004v^11 - 355005349v^9 + 691228436v^7 + 2139497850v^5 + 3370777175v^3 + 3295922875v) / 858900625
$\beta_{14}$	$=$	$ ( - 2000435 \nu^{15} + 181037 \nu^{13} + 8319942 \nu^{11} + 89497307 \nu^{9} + \cdots - 651093200 \nu ) / 171780125 $ (-2000435v^15 + 181037v^13 + 8319942v^11 + 89497307v^9 - 126320558v^7 - 670609003v^5 - 1070064880v^3 - 651093200v) / 171780125
$\beta_{15}$	$=$	$ ( - 14757833 \nu^{15} + 273347 \nu^{13} + 57022912 \nu^{11} + 679004122 \nu^{9} + \cdots - 6958879375 \nu ) / 858900625 $ (-14757833v^15 + 273347v^13 + 57022912v^11 + 679004122v^9 - 877490958v^7 - 4930710405v^5 - 8853033325v^3 - 6958879375v) / 858900625

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

$\nu$	$=$	$ ( -\beta_{15} + 2\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + 2\beta_{10} + 3\beta_{9} + 2\beta_{5} ) / 5 $ (-b15 + 2b14 - b13 - b12 - b11 + 2b10 + 3b9 + 2b5) / 5
$\nu^{2}$	$=$	$ ( -2\beta_{8} + 2\beta_{7} + 3\beta_{6} + 2\beta_{4} + 11\beta_{3} - 4\beta_{2} - \beta _1 + 4 ) / 5 $ (-2b8 + 2b7 + 3b6 + 2b4 + 11b3 - 4b2 - b1 + 4) / 5
$\nu^{3}$	$=$	$ ( -\beta_{15} - 3\beta_{14} - 16\beta_{13} - 6\beta_{12} - \beta_{11} + 17\beta_{10} - 2\beta_{9} + 2\beta_{5} ) / 5 $ (-b15 - 3b14 - 16b13 - 6b12 - b11 + 17b10 - 2b9 + 2b5) / 5
$\nu^{4}$	$=$	$ ( -7\beta_{8} - 13\beta_{7} + 3\beta_{6} + 7\beta_{4} + 6\beta_{3} - 19\beta_{2} - 26\beta _1 + 14 ) / 5 $ (-7b8 - 13b7 + 3b6 + 7b4 + 6b3 - 19b2 - 26*b1 + 14) / 5
$\nu^{5}$	$=$	$ ( -16\beta_{15} + 22\beta_{14} - 6\beta_{13} - 51\beta_{12} - 6\beta_{11} - 43\beta_{10} - 67\beta_{9} - 8\beta_{5} ) / 5 $ (-16b15 + 22b14 - 6b13 - 51b12 - 6b11 - 43b10 - 67b9 - 8b5) / 5
$\nu^{6}$	$=$	$ ( 23\beta_{8} - 23\beta_{7} + 63\beta_{6} + 52\beta_{4} + 156\beta_{3} + 11\beta_{2} - 11\beta _1 + 144 ) / 5 $ (23b8 - 23b7 + 63b6 + 52b4 + 156b3 + 11b2 - 11*b1 + 144) / 5
$\nu^{7}$	$=$	$ ( - 31 \beta_{15} - 53 \beta_{14} + 9 \beta_{13} - 31 \beta_{12} - 106 \beta_{11} - 33 \beta_{10} + \cdots + 137 \beta_{5} ) / 5 $ (-31b15 - 53b14 + 9b13 - 31b12 - 106b11 - 33b10 - 62b9 + 137b5) / 5
$\nu^{8}$	$=$	$ ( -57\beta_{8} - 18\beta_{7} + 363\beta_{6} - 208\beta_{4} + 381\beta_{3} - 114\beta_{2} - 96\beta _1 + 39 ) / 5 $ (-57b8 - 18b7 + 363b6 - 208b4 + 381b3 - 114b2 - 96*b1 + 39) / 5
$\nu^{9}$	$=$	$ ( 209 \beta_{15} - 418 \beta_{14} - 271 \beta_{13} - 551 \beta_{12} - 76 \beta_{11} - 133 \beta_{10} + \cdots + 342 \beta_{5} ) / 5 $ (209b15 - 418b14 - 271b13 - 551b12 - 76b11 - 133b10 - 627b9 + 342b5) / 5
$\nu^{10}$	$=$	$ ( 208\beta_{8} - 688\beta_{7} + 208\beta_{6} - 493\beta_{4} - 149\beta_{3} - 344\beta_{2} - 896\beta _1 - 606 ) / 5 $ (208b8 - 688b7 + 208b6 - 493b4 - 149b3 - 344b2 - 896*b1 - 606) / 5
$\nu^{11}$	$=$	$ ( - 136 \beta_{15} + 77 \beta_{14} + 1584 \beta_{13} - 1661 \beta_{12} - 136 \beta_{11} + \cdots + 77 \beta_{5} ) / 5 $ (-136b15 + 77b14 + 1584b13 - 1661b12 - 136b11 - 4208b10 - 4267b9 + 77b5) / 5
$\nu^{12}$	$=$	$ ( 2208 \beta_{8} - 683 \beta_{7} + 2888 \beta_{6} - 683 \beta_{4} + 3956 \beta_{3} + 2891 \beta_{2} + \cdots + 1784 ) / 5 $ (2208b8 - 683b7 + 2888b6 - 683b4 + 3956b3 + 2891b2 + 1104*b1 + 1784) / 5
$\nu^{13}$	$=$	$ ( 1299 \beta_{15} - 5453 \beta_{14} + 5559 \beta_{13} + 2144 \beta_{12} - 3376 \beta_{11} + \cdots + 6752 \beta_{5} ) / 5 $ (1299b15 - 5453b14 + 5559b13 + 2144b12 - 3376b11 - 6858b10 - 2077b9 + 6752b5) / 5
$\nu^{14}$	$=$	$ ( 393 \beta_{8} + 1012 \beta_{7} + 10778 \beta_{6} - 18118 \beta_{4} - 619 \beta_{3} + 1631 \beta_{2} + \cdots - 18511 ) / 5 $ (393b8 + 1012b7 + 10778b6 - 18118b4 - 619b3 + 1631b2 + 2024*b1 - 18511) / 5
$\nu^{15}$	$=$	$ ( 17084 \beta_{15} - 22378 \beta_{14} + 5294 \beta_{13} - 8776 \beta_{12} + 5294 \beta_{11} + \cdots + 8542 \beta_{5} ) / 5 $ (17084b15 - 22378b14 + 5294b13 - 8776b12 + 5294b11 - 17318b10 - 19517b9 + 8542b5) / 5

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/250\mathbb{Z}\right)^\times$.

$n$	$127$
$\chi(n)$	$-\beta_{3}$

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} $

49.1

0.917186	+	1.66637i
−1.86824	−	0.357358i
1.86824	+	0.357358i
−0.917186	−	1.66637i
0.644389	+	0.983224i
−0.0566033	−	1.17421i
0.0566033	+	1.17421i
−0.644389	−	0.983224i
0.644389	−	0.983224i
−0.0566033	+	1.17421i
0.0566033	−	1.17421i
−0.644389	+	0.983224i
0.917186	−	1.66637i
−1.86824	+	0.357358i
1.86824	−	0.357358i
−0.917186	+	1.66637i

−0.587785

0.809017i

−1.63079

0.529876i

−0.309017

−

0.951057i

0.529876

−

1.63079i

2.77447i

0.951057

0.309017i

−0.0483405

0.0351215i

49.2

−0.587785

0.809017i

2.21858

−

0.720859i

−0.309017

−

0.951057i

−0.720859

2.21858i

−

3.77447i

0.951057

0.309017i

1.97539

−

1.43521i

49.3

0.587785

−

0.809017i

−2.21858

0.720859i

−0.309017

−

0.951057i

−0.720859

2.21858i

3.77447i

−0.951057

−

0.309017i

1.97539

−

1.43521i

49.4

0.587785

−

0.809017i

1.63079

−

0.529876i

−0.309017

−

0.951057i

0.529876

−

1.63079i

−

2.77447i

−0.951057

−

0.309017i

−0.0483405

0.0351215i

99.1

−0.951057

0.309017i

−0.792578

−

1.09089i

0.809017

−

0.587785i

1.09089

0.792578i

−

0.833366i

−0.587785

0.809017i

0.365190

−

1.12394i

99.2

−0.951057

0.309017i

1.74363

2.39991i

0.809017

−

0.587785i

−2.39991

−

1.74363i

1.83337i

−0.587785

0.809017i

−1.79224

5.51595i

99.3

0.951057

−

0.309017i

−1.74363

−

2.39991i

0.809017

−

0.587785i

−2.39991

−

1.74363i

−

1.83337i

0.587785

−

0.809017i

−1.79224

5.51595i

99.4

0.951057

−

0.309017i

0.792578

1.09089i

0.809017

−

0.587785i

1.09089

0.792578i

0.833366i

0.587785

−

0.809017i

0.365190

−

1.12394i

149.1

−0.951057

−

0.309017i

−0.792578

1.09089i

0.809017

0.587785i

1.09089

−

0.792578i

0.833366i

−0.587785

−

0.809017i

0.365190

1.12394i

149.2

−0.951057

−

0.309017i

1.74363

−

2.39991i

0.809017

0.587785i

−2.39991

1.74363i

−

1.83337i

−0.587785

−

0.809017i

−1.79224

−

5.51595i

149.3

0.951057

0.309017i

−1.74363

2.39991i

0.809017

0.587785i

−2.39991

1.74363i

1.83337i

0.587785

0.809017i

−1.79224

−

5.51595i

149.4

0.951057

0.309017i

0.792578

−

1.09089i

0.809017

0.587785i

1.09089

−

0.792578i

−

0.833366i

0.587785

0.809017i

0.365190

1.12394i

199.1

−0.587785

−

0.809017i

−1.63079

−

0.529876i

−0.309017

0.951057i

0.529876

1.63079i

−

2.77447i

0.951057

−

0.309017i

−0.0483405

−

0.0351215i

199.2

−0.587785

−

0.809017i

2.21858

0.720859i

−0.309017

0.951057i

−0.720859

−

2.21858i

3.77447i

0.951057

−

0.309017i

1.97539

1.43521i

199.3

0.587785

0.809017i

−2.21858

−

0.720859i

−0.309017

0.951057i

−0.720859

−

2.21858i

−

3.77447i

−0.951057

0.309017i

1.97539

1.43521i

199.4

0.587785

0.809017i

1.63079

0.529876i

−0.309017

0.951057i

0.529876

1.63079i

2.77447i

−0.951057

0.309017i

−0.0483405

−

0.0351215i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
25.d	even	5	1	inner
25.e	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	250.2.e.c		16
5.b	even	2	1	inner	250.2.e.c		16
5.c	odd	4	1		50.2.d.b	✓	8
5.c	odd	4	1		250.2.d.d		8
15.e	even	4	1		450.2.h.e		8
20.e	even	4	1		400.2.u.d		8
25.d	even	5	1	inner	250.2.e.c		16
25.d	even	5	1		1250.2.b.e		8
25.e	even	10	1	inner	250.2.e.c		16
25.e	even	10	1		1250.2.b.e		8
25.f	odd	20	1		50.2.d.b	✓	8
25.f	odd	20	1		250.2.d.d		8
25.f	odd	20	1		1250.2.a.f		4
25.f	odd	20	1		1250.2.a.l		4
75.l	even	20	1		450.2.h.e		8
100.l	even	20	1		400.2.u.d		8
100.l	even	20	1		10000.2.a.t		4
100.l	even	20	1		10000.2.a.x		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
50.2.d.b	✓	8	5.c	odd	4	1
50.2.d.b	✓	8	25.f	odd	20	1
250.2.d.d		8	5.c	odd	4	1
250.2.d.d		8	25.f	odd	20	1
250.2.e.c		16	1.a	even	1	1	trivial
250.2.e.c		16	5.b	even	2	1	inner
250.2.e.c		16	25.d	even	5	1	inner
250.2.e.c		16	25.e	even	10	1	inner
400.2.u.d		8	20.e	even	4	1
400.2.u.d		8	100.l	even	20	1
450.2.h.e		8	15.e	even	4	1
450.2.h.e		8	75.l	even	20	1
1250.2.a.f		4	25.f	odd	20	1
1250.2.a.l		4	25.f	odd	20	1
1250.2.b.e		8	25.d	even	5	1
1250.2.b.e		8	25.e	even	10	1
10000.2.a.t		4	100.l	even	20	1
10000.2.a.x		4	100.l	even	20	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{16} - 7T_{3}^{14} + 78T_{3}^{12} - 764T_{3}^{10} + 4625T_{3}^{8} - 12224T_{3}^{6} + 19968T_{3}^{4} - 28672T_{3}^{2} + 65536 $ T3^16 - 7*T3^14 + 78*T3^12 - 764*T3^10 + 4625*T3^8 - 12224*T3^6 + 19968*T3^4 - 28672*T3^2 + 65536 acting on $S_{2}^{\mathrm{new}}(250, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} $ (T^8 - T^6 + T^4 - T^2 + 1)^2
$3$	$ T^{16} - 7 T^{14} + \cdots + 65536 $ T^16 - 7T^14 + 78T^12 - 764T^10 + 4625T^8 - 12224T^6 + 19968T^4 - 28672*T^2 + 65536
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} + 26 T^{6} + \cdots + 256)^{2} $ (T^8 + 26T^6 + 201T^4 + 496*T^2 + 256)^2
$11$	$ (T^{8} - T^{7} - 3 T^{6} + \cdots + 256)^{2} $ (T^8 - T^7 - 3T^6 + 7T^5 + 105T^4 + 128T^3 + 352T^2 - 64T + 256)^2
$13$	$ T^{16} + \cdots + 1568239201 $ T^16 - 17T^14 + 923T^12 - 30774T^10 + 611375T^8 - 4132974T^6 + 57271058T^4 - 429750052*T^2 + 1568239201
$17$	$ T^{16} + \cdots + 141158161 $ T^16 - 23T^14 + 808T^12 - 25061T^10 + 626475T^8 - 330601T^6 + 34950648T^4 - 112192283*T^2 + 141158161
$19$	$ (T^{8} + 20 T^{7} + \cdots + 6400)^{2} $ (T^8 + 20T^7 + 190T^6 + 965T^5 + 3585T^4 + 8000T^3 + 12000T^2 + 11200*T + 6400)^2
$23$	$ T^{16} - 117 T^{14} + \cdots + 65536 $ T^16 - 117T^14 + 5673T^12 - 84269T^10 + 503505T^8 + 344896T^6 + 511488T^4 + 135168*T^2 + 65536
$29$	$ (T^{8} - 15 T^{7} + \cdots + 25)^{2} $ (T^8 - 15T^7 + 105T^6 - 390T^5 + 1735T^4 - 5700T^3 + 10150T^2 + 300*T + 25)^2
$31$	$ (T^{8} + 9 T^{7} + \cdots + 1597696)^{2} $ (T^8 + 9T^7 + 87T^6 + 237T^5 + 805T^4 + 228T^3 + 56432T^2 + 257856*T + 1597696)^2
$37$	$ T^{16} - 58 T^{14} + \cdots + 25411681 $ T^16 - 58T^14 + 2793T^12 - 131476T^10 + 7335030T^8 - 15259546T^6 + 16392788T^4 - 9920688*T^2 + 25411681
$41$	$ (T^{8} + 9 T^{7} + \cdots + 7921)^{2} $ (T^8 + 9T^7 + 27T^6 - 128T^5 + 1425T^4 - 4232T^3 + 9162T^2 - 11214*T + 7921)^2
$43$	$ (T^{8} + 114 T^{6} + \cdots + 30976)^{2} $ (T^8 + 114T^6 + 2641T^4 + 17824*T^2 + 30976)^2
$47$	$ T^{16} - 33 T^{14} + \cdots + 65536 $ T^16 - 33T^14 + 473T^12 - 2961T^10 + 9025T^8 - 16176T^6 + 132608T^4 + 49152*T^2 + 65536
$53$	$ T^{16} + \cdots + 4294967296 $ T^16 - 167T^14 + 11158T^12 + 115916T^10 + 59314305T^8 + 955607296T^6 + 55624204288T^4 - 25031606272*T^2 + 4294967296
$59$	$ (T^{8} + 10 T^{7} + \cdots + 102400)^{2} $ (T^8 + 10T^7 + 260T^6 + 1320T^5 + 17360T^4 + 105600T^3 + 320000T^2 + 281600*T + 102400)^2
$61$	$ (T^{8} - 6 T^{7} + \cdots + 2920681)^{2} $ (T^8 - 6T^7 - 18T^6 + 332T^5 + 9180T^4 - 56572T^3 + 570837T^2 - 984384*T + 2920681)^2
$67$	$ T^{16} + \cdots + 794123370496 $ T^16 - 353T^14 + 51753T^12 - 1334641T^10 + 462689105T^8 - 12950447296T^6 + 140360683008T^4 + 149382909952*T^2 + 794123370496
$71$	$ (T^{8} + 9 T^{7} + \cdots + 4096)^{2} $ (T^8 + 9T^7 + 217T^6 + 297T^5 + 5225T^4 - 21432T^3 + 32512T^2 + 19456*T + 4096)^2
$73$	$ T^{16} + \cdots + 1380756603136 $ T^16 - 352T^14 + 47318T^12 + 167361T^10 + 43986725T^8 - 317475324T^6 + 7889246048T^4 - 131455864832*T^2 + 1380756603136
$79$	$ (T^{8} - 10 T^{7} + \cdots + 102400)^{2} $ (T^8 - 10T^7 + 120T^6 - 320T^5 - 240T^4 + 3200T^3 + 25600T^2 - 25600*T + 102400)^2
$83$	$ T^{16} + \cdots + 180227832610816 $ T^16 - 77T^14 + 19833T^12 - 3107189T^10 + 283380705T^8 - 6695212304T^6 + 426901737728T^4 - 12797040463872*T^2 + 180227832610816
$89$	$ (T^{8} - 15 T^{7} + \cdots + 9610000)^{2} $ (T^8 - 15T^7 + 70T^6 + 750T^5 + 8525T^4 - 30750T^3 + 1025500T^2 - 2015000*T + 9610000)^2
$97$	$ (T^{8} + 76 T^{6} + 15126 T^{4} + \cdots + 1)^{2} $ (T^8 + 76T^6 + 15126T^4 - 199*T^2 + 1)^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:	\( N \)	\(=\)	\( 250 = 2 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	250.e (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{13} + \beta_{12} + \cdots + \beta_{9}) q^{2} + ( - \beta_{14} + \beta_{13} - \beta_{9}) q^{3} - \beta_{3} q^{4} + (\beta_{8} + \beta_{6} + \cdots + \beta_{3}) q^{6} + ( - \beta_{14} - \beta_{11} + \cdots + \beta_{5}) q^{7}+ \cdots + ( - 2 \beta_{8} - \beta_{4} - \beta_{3} + \cdots + 6) q^{99}+O(q^{100}) \) q + (-b13 + b12 + b10 + b9) * q^2 + (-b14 + b13 - b9) * q^3 - b3 * q^4 + (b8 + b6 + b4 + b3) * q^6 + (-b14 - b11 - b9 + b5) * q^7 + b12 * q^8 + (-b8 - b7 - 3b6 - b4 - b2 - 1) q^9 + (b6 + b4 + b3 - b2 - b1 + 1) * q^11 + (-b15 + b13 - b12 - b9) * q^12 + (-b15 - b13 - 2b12 + b11 - 2b9 - b5) * q^13 + (b6 + b4 + b3 - b2 - b1 + 1) * q^14 + b6 * q^16 + (b15 - b14 + b13 - 3b12 - b10) q^17 + (b14 - b12 + b11 - b10 + 2b9) q^18 + (-b8 - b7 + 3b6 + 2b4 + 3b3 - b1) q^19 + (-5b6 - 4b3 + b1 - 5) * q^21 + (-b15 + b14 + b10 - b5) * q^22 + (2b15 - b14 - 3b13 + 3b12 - 2b11 + b10 + b9 + b5) * q^23 + (b4 + b3 - b2) * q^24 + (b8 + 2b4 + 2b3 + b1 + 3) * q^26 + (b14 - 5b13 + 5b12 + b10 + b9 - b5) * q^27 + (-b15 + b14 + b10 - b5) * q^28 + (2b6 - 3b3 - b1 + 2) * q^29 + (-2b8 - b7 + 2b6 - 3b4 + 2b3 - b1) * q^31 - b9 * q^32 + (5b13 - b12 + b11 - 5b10) * q^33 + (b8 - 3b6 - b4 + b2 - 1) q^34 + (-b8 - b7 - 3b6 - 3b4 - 2b3 - 2) q^36 + (2b15 + 2b13 + b12 + b9) * q^37 + (b15 + 2b13 - 3b12 - b11 - 3b9 + b5) q^38 + (b8 + b7 - 5b6 - 5b4 - 2b3 + 3b2 + 3b1 - 2) q^39 + (-b8 + b7 + 3b6 - b2) q^41 + (5b13 - b12 + b11 - 5b10) * q^42 + (-b14 + 4b12 - b11 + 4b10 + 3b9 + b5) q^43 + (b7 + b4 + b1) * q^44 + (b7 + 2b6 - b3 + b2 - b1 + 2) q^46 + (b15 - b14 - b10 + b5) * q^47 + (-b15 + b14 + b13 - b12 + b11 - b5) * q^48 + (b8 - 4b4 - 4b3 + b2 + b1 - 2) * q^49 + (-b8 + b4 + b3 + 2b2 - b1 + 5) q^51 + (-b15 - b13 + b12 + b11 + 3b10 + 3b9) * q^52 + (3b14 - 4b13 + 4b10 + 4b9) * q^53 + (b7 + 4b6 - b3 + b2 + b1 + 4) q^54 + (b7 + b4 + b1) * q^56 + (3b14 + 2b12 + 3b11 + 2b10 + 3b9 - b5) q^57 + (-2b13 + 5b12 - b11 + 2b10) q^58 + (2b8 + 4b7 + 2b6 + 2b2) * q^59 + (2b8 + 2b7 - 3b6 - 3b4 - 2b3 + 4b2 + 4b1 - 2) q^61 + (2b15 - 3b13 - 2b12 - b11 - 2b9 + b5) * q^62 + (b15 - 7b13 + b12 - 2b11 + b9 + 2b5) q^63 + (b6 + b4 + b3 + 1) * q^64 + (-b7 - b6 - 5b4 - 5) q^66 + (-3b15 + 3b14 - 7b12) q^67 + (-b14 - b12 - b11 - b10 + 2b9 + b5) q^68 + (3b8 + 2b7 + 7b6 + 13b4 + 7b3 + 2b1) * q^69 + (-b7 - 4b6 - 7b3 - b2 - b1 - 4) * q^71 + (b15 - b13 - 2b10 + b9 + b5) q^72 + (-3b15 + b14 + 8b13 - 8b12 + 3b11 - 6b10 - 6b9 - b5) * q^73 + (-b4 - b3 + 2b2 - 3) q^74 + (-b8 + 3b4 + 3b3 - b1 + 1) * q^76 + (-b14 + 5b13 - 5b12 - 9b10 - 9b9 + b5) * q^77 + (2b15 - 3b14 - 3b13 - 2b10 + 3b9 + 2b5) * q^78 + (2b6 + 4b3 + 2b1 + 2) q^79 + (2b8 + b7 + 6b6 + 6b3 + b1) q^81 + (b14 + b11 - 3b9 - 2b5) * q^82 + (-b15 + b14 + 2b13 - 7b12 - 2b11 - 2b10) * q^83 + (-b7 - b6 - 5b4 - 5) q^84 + (b6 + b4 - 3b3 - b2 - b1 - 3) q^86 + (-5b15 + 11b13 - 5b12 + 2b11 - 5b9 - 2b5) * q^87 + (b13 + b11 - b5) * q^88 + (-b8 - b7 - 2b6 - 2b4 + 2b3 - 4b2 - 4b1 + 2) q^89 + (3b8 - 2b7 - 3b6 - 2b4 + 3b2 - 2) q^91 + (b15 - b14 - 2b13 + 3b12 - 2b11 + 2b10) * q^92 + (2b14 + 7b12 + 2b11 + 7b10 - 2b9 - 5b5) * q^93 + (-b7 - b4 - b1) * q^94 + (-b6 + b1 - 1) * q^96 + (-5b13 - 3b10 + 5b9) q^97 + (-b14 - 2b13 + 2b12 - 2b10 - 2b9 + b5) * q^98 + (-2b8 - b4 - b3 - b2 - 2b1 + 6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 2 q^{11} + 2 q^{14} - 4 q^{16} - 40 q^{19} - 38 q^{21} - 4 q^{24} + 44 q^{26} + 30 q^{29} - 18 q^{31} + 2 q^{34} - 2 q^{36} + 24 q^{39} - 18 q^{41} - 2 q^{44} + 14 q^{46}+ \cdots + 84 q^{99}+O(q^{100}) \) 16 * q + 4 * q^4 - 6 * q^6 + 2 * q^9 + 2 * q^11 + 2 * q^14 - 4 * q^16 - 40 * q^19 - 38 * q^21 - 4 * q^24 + 44 * q^26 + 30 * q^29 - 18 * q^31 + 2 * q^34 - 2 * q^36 + 24 * q^39 - 18 * q^41 - 2 * q^44 + 14 * q^46 + 8 * q^49 + 52 * q^51 + 50 * q^54 - 2 * q^56 - 20 * q^59 + 12 * q^61 + 4 * q^64 - 52 * q^66 - 86 * q^69 - 18 * q^71 - 48 * q^74 - 20 * q^76 + 20 * q^79 - 34 * q^81 - 52 * q^84 - 46 * q^86 + 30 * q^89 + 2 * q^91 + 2 * q^94 - 6 * q^96 + 84 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	250.2.e.c

Level	$250$
Weight	$2$
Character orbit	250.e
Analytic conductor	$1.996$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.99626005053\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) x^16 + x^14 - 4x^12 - 49x^10 + 11x^8 + 395x^6 + 900x^4 + 1125x^2 + 625
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 5^{2} \)
Twist minimal:	no (minimal twist has level 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(\beta_{1}\)	\(=\)	\( ( 948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + \cdots + 181387875 ) / 171780125 \) (948392v^14 - 2081693v^12 - 1547103v^10 - 35207443v^8 + 136185777v^6 + 77053830v^4 + 27131400*v^2 + 181387875) / 171780125
\(\beta_{2}\)	\(=\)	\( ( 122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + \cdots + 52412250 ) / 15616375 \) (122987v^14 + 97172v^12 - 701513v^10 - 5799603v^8 + 3932417v^6 + 54634025v^4 + 77779875*v^2 + 52412250) / 15616375
\(\beta_{3}\)	\(=\)	\( ( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + \cdots + 563878625 ) / 171780125 \) (1379032v^14 - 312743v^12 - 4990978v^10 - 61900293v^8 + 94590252v^6 + 424939915v^4 + 771306350*v^2 + 563878625) / 171780125
\(\beta_{4}\)	\(=\)	\( ( - 281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} + \cdots - 167096475 ) / 34356025 \) (-281280v^14 + 69219v^12 + 939009v^10 + 12381889v^8 - 18275316v^6 - 83450271v^4 - 152170830*v^2 - 167096475) / 34356025
\(\beta_{5}\)	\(=\)	\( ( - 1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} + \cdots + 1705295750 \nu ) / 858900625 \) (-1157122v^15 + 1428203v^13 + 8678488v^11 + 43919978v^9 - 137248467v^7 - 474206065v^5 + 222668250v^3 + 1705295750v) / 858900625
\(\beta_{6}\)	\(=\)	\( ( - 441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} + \cdots - 215410900 ) / 34356025 \) (-441862v^14 + 108648v^12 + 1544473v^10 + 20140738v^8 - 29602957v^6 - 135351515v^4 - 246968035*v^2 - 215410900) / 34356025
\(\beta_{7}\)	\(=\)	\( ( - 2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} + \cdots - 860602375 ) / 171780125 \) (-2817591v^14 + 2379174v^12 + 8884104v^10 + 118381374v^8 - 262226761v^6 - 716617430v^4 - 884417625*v^2 - 860602375) / 171780125
\(\beta_{8}\)	\(=\)	\( ( - 626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} + \cdots - 222107450 ) / 34356025 \) (-626638v^14 + 144621v^12 + 2783831v^10 + 27217946v^8 - 41997039v^6 - 210462216v^4 - 281904075*v^2 - 222107450) / 34356025
\(\beta_{9}\)	\(=\)	\( ( 2293 \nu^{15} + 486 \nu^{13} - 12284 \nu^{11} - 101834 \nu^{9} + 117086 \nu^{7} + 906528 \nu^{5} + \cdots + 955700 \nu ) / 633875 \) (2293v^15 + 486v^13 - 12284v^11 - 101834v^9 + 117086v^7 + 906528v^5 + 1102430v^3 + 955700v) / 633875
\(\beta_{10}\)	\(=\)	\( ( 3183877 \nu^{15} - 4006568 \nu^{13} - 6543803 \nu^{11} - 134015793 \nu^{9} + \cdots + 1245786250 \nu ) / 858900625 \) (3183877v^15 - 4006568v^13 - 6543803v^11 - 134015793v^9 + 335534227v^7 + 577777045v^5 + 1177699300v^3 + 1245786250v) / 858900625
\(\beta_{11}\)	\(=\)	\( ( 8105189 \nu^{15} + 777904 \nu^{13} - 26062291 \nu^{11} - 376639746 \nu^{9} + \cdots + 5975622625 \nu ) / 858900625 \) (8105189v^15 + 777904v^13 - 26062291v^11 - 376639746v^9 + 389714419v^7 + 2537763645v^5 + 5645322900v^3 + 5975622625v) / 858900625
\(\beta_{12}\)	\(=\)	\( ( - 8189122 \nu^{15} + 3158678 \nu^{13} + 32153713 \nu^{11} + 353467203 \nu^{9} + \cdots - 3331016750 \nu ) / 858900625 \) (-8189122v^15 + 3158678v^13 + 32153713v^11 + 353467203v^9 - 594131367v^7 - 2560462840v^5 - 3581602500v^3 - 3331016750v) / 858900625
\(\beta_{13}\)	\(=\)	\( ( 8211971 \nu^{15} - 5580699 \nu^{13} - 25580004 \nu^{11} - 355005349 \nu^{9} + \cdots + 3295922875 \nu ) / 858900625 \) (8211971v^15 - 5580699v^13 - 25580004v^11 - 355005349v^9 + 691228436v^7 + 2139497850v^5 + 3370777175v^3 + 3295922875v) / 858900625
\(\beta_{14}\)	\(=\)	\( ( - 2000435 \nu^{15} + 181037 \nu^{13} + 8319942 \nu^{11} + 89497307 \nu^{9} + \cdots - 651093200 \nu ) / 171780125 \) (-2000435v^15 + 181037v^13 + 8319942v^11 + 89497307v^9 - 126320558v^7 - 670609003v^5 - 1070064880v^3 - 651093200v) / 171780125
\(\beta_{15}\)	\(=\)	\( ( - 14757833 \nu^{15} + 273347 \nu^{13} + 57022912 \nu^{11} + 679004122 \nu^{9} + \cdots - 6958879375 \nu ) / 858900625 \) (-14757833v^15 + 273347v^13 + 57022912v^11 + 679004122v^9 - 877490958v^7 - 4930710405v^5 - 8853033325v^3 - 6958879375v) / 858900625

\(\nu\)	\(=\)	\( ( -\beta_{15} + 2\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + 2\beta_{10} + 3\beta_{9} + 2\beta_{5} ) / 5 \) (-b15 + 2b14 - b13 - b12 - b11 + 2b10 + 3b9 + 2b5) / 5
\(\nu^{2}\)	\(=\)	\( ( -2\beta_{8} + 2\beta_{7} + 3\beta_{6} + 2\beta_{4} + 11\beta_{3} - 4\beta_{2} - \beta _1 + 4 ) / 5 \) (-2b8 + 2b7 + 3b6 + 2b4 + 11b3 - 4b2 - b1 + 4) / 5
\(\nu^{3}\)	\(=\)	\( ( -\beta_{15} - 3\beta_{14} - 16\beta_{13} - 6\beta_{12} - \beta_{11} + 17\beta_{10} - 2\beta_{9} + 2\beta_{5} ) / 5 \) (-b15 - 3b14 - 16b13 - 6b12 - b11 + 17b10 - 2b9 + 2b5) / 5
\(\nu^{4}\)	\(=\)	\( ( -7\beta_{8} - 13\beta_{7} + 3\beta_{6} + 7\beta_{4} + 6\beta_{3} - 19\beta_{2} - 26\beta _1 + 14 ) / 5 \) (-7b8 - 13b7 + 3b6 + 7b4 + 6b3 - 19b2 - 26*b1 + 14) / 5
\(\nu^{5}\)	\(=\)	\( ( -16\beta_{15} + 22\beta_{14} - 6\beta_{13} - 51\beta_{12} - 6\beta_{11} - 43\beta_{10} - 67\beta_{9} - 8\beta_{5} ) / 5 \) (-16b15 + 22b14 - 6b13 - 51b12 - 6b11 - 43b10 - 67b9 - 8b5) / 5
\(\nu^{6}\)	\(=\)	\( ( 23\beta_{8} - 23\beta_{7} + 63\beta_{6} + 52\beta_{4} + 156\beta_{3} + 11\beta_{2} - 11\beta _1 + 144 ) / 5 \) (23b8 - 23b7 + 63b6 + 52b4 + 156b3 + 11b2 - 11*b1 + 144) / 5
\(\nu^{7}\)	\(=\)	\( ( - 31 \beta_{15} - 53 \beta_{14} + 9 \beta_{13} - 31 \beta_{12} - 106 \beta_{11} - 33 \beta_{10} + \cdots + 137 \beta_{5} ) / 5 \) (-31b15 - 53b14 + 9b13 - 31b12 - 106b11 - 33b10 - 62b9 + 137b5) / 5
\(\nu^{8}\)	\(=\)	\( ( -57\beta_{8} - 18\beta_{7} + 363\beta_{6} - 208\beta_{4} + 381\beta_{3} - 114\beta_{2} - 96\beta _1 + 39 ) / 5 \) (-57b8 - 18b7 + 363b6 - 208b4 + 381b3 - 114b2 - 96*b1 + 39) / 5
\(\nu^{9}\)	\(=\)	\( ( 209 \beta_{15} - 418 \beta_{14} - 271 \beta_{13} - 551 \beta_{12} - 76 \beta_{11} - 133 \beta_{10} + \cdots + 342 \beta_{5} ) / 5 \) (209b15 - 418b14 - 271b13 - 551b12 - 76b11 - 133b10 - 627b9 + 342b5) / 5
\(\nu^{10}\)	\(=\)	\( ( 208\beta_{8} - 688\beta_{7} + 208\beta_{6} - 493\beta_{4} - 149\beta_{3} - 344\beta_{2} - 896\beta _1 - 606 ) / 5 \) (208b8 - 688b7 + 208b6 - 493b4 - 149b3 - 344b2 - 896*b1 - 606) / 5
\(\nu^{11}\)	\(=\)	\( ( - 136 \beta_{15} + 77 \beta_{14} + 1584 \beta_{13} - 1661 \beta_{12} - 136 \beta_{11} + \cdots + 77 \beta_{5} ) / 5 \) (-136b15 + 77b14 + 1584b13 - 1661b12 - 136b11 - 4208b10 - 4267b9 + 77b5) / 5
\(\nu^{12}\)	\(=\)	\( ( 2208 \beta_{8} - 683 \beta_{7} + 2888 \beta_{6} - 683 \beta_{4} + 3956 \beta_{3} + 2891 \beta_{2} + \cdots + 1784 ) / 5 \) (2208b8 - 683b7 + 2888b6 - 683b4 + 3956b3 + 2891b2 + 1104*b1 + 1784) / 5
\(\nu^{13}\)	\(=\)	\( ( 1299 \beta_{15} - 5453 \beta_{14} + 5559 \beta_{13} + 2144 \beta_{12} - 3376 \beta_{11} + \cdots + 6752 \beta_{5} ) / 5 \) (1299b15 - 5453b14 + 5559b13 + 2144b12 - 3376b11 - 6858b10 - 2077b9 + 6752b5) / 5
\(\nu^{14}\)	\(=\)	\( ( 393 \beta_{8} + 1012 \beta_{7} + 10778 \beta_{6} - 18118 \beta_{4} - 619 \beta_{3} + 1631 \beta_{2} + \cdots - 18511 ) / 5 \) (393b8 + 1012b7 + 10778b6 - 18118b4 - 619b3 + 1631b2 + 2024*b1 - 18511) / 5
\(\nu^{15}\)	\(=\)	\( ( 17084 \beta_{15} - 22378 \beta_{14} + 5294 \beta_{13} - 8776 \beta_{12} + 5294 \beta_{11} + \cdots + 8542 \beta_{5} ) / 5 \) (17084b15 - 22378b14 + 5294b13 - 8776b12 + 5294b11 - 17318b10 - 19517b9 + 8542b5) / 5

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) (T^8 - T^6 + T^4 - T^2 + 1)^2
$3$	\( T^{16} - 7 T^{14} + \cdots + 65536 \) T^16 - 7T^14 + 78T^12 - 764T^10 + 4625T^8 - 12224T^6 + 19968T^4 - 28672*T^2 + 65536
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} + 26 T^{6} + \cdots + 256)^{2} \) (T^8 + 26T^6 + 201T^4 + 496*T^2 + 256)^2
$11$	\( (T^{8} - T^{7} - 3 T^{6} + \cdots + 256)^{2} \) (T^8 - T^7 - 3T^6 + 7T^5 + 105T^4 + 128T^3 + 352T^2 - 64T + 256)^2
$13$	\( T^{16} + \cdots + 1568239201 \) T^16 - 17T^14 + 923T^12 - 30774T^10 + 611375T^8 - 4132974T^6 + 57271058T^4 - 429750052*T^2 + 1568239201
$17$	\( T^{16} + \cdots + 141158161 \) T^16 - 23T^14 + 808T^12 - 25061T^10 + 626475T^8 - 330601T^6 + 34950648T^4 - 112192283*T^2 + 141158161
$19$	\( (T^{8} + 20 T^{7} + \cdots + 6400)^{2} \) (T^8 + 20T^7 + 190T^6 + 965T^5 + 3585T^4 + 8000T^3 + 12000T^2 + 11200*T + 6400)^2
$23$	\( T^{16} - 117 T^{14} + \cdots + 65536 \) T^16 - 117T^14 + 5673T^12 - 84269T^10 + 503505T^8 + 344896T^6 + 511488T^4 + 135168*T^2 + 65536
$29$	\( (T^{8} - 15 T^{7} + \cdots + 25)^{2} \) (T^8 - 15T^7 + 105T^6 - 390T^5 + 1735T^4 - 5700T^3 + 10150T^2 + 300*T + 25)^2
$31$	\( (T^{8} + 9 T^{7} + \cdots + 1597696)^{2} \) (T^8 + 9T^7 + 87T^6 + 237T^5 + 805T^4 + 228T^3 + 56432T^2 + 257856*T + 1597696)^2
$37$	\( T^{16} - 58 T^{14} + \cdots + 25411681 \) T^16 - 58T^14 + 2793T^12 - 131476T^10 + 7335030T^8 - 15259546T^6 + 16392788T^4 - 9920688*T^2 + 25411681
$41$	\( (T^{8} + 9 T^{7} + \cdots + 7921)^{2} \) (T^8 + 9T^7 + 27T^6 - 128T^5 + 1425T^4 - 4232T^3 + 9162T^2 - 11214*T + 7921)^2
$43$	\( (T^{8} + 114 T^{6} + \cdots + 30976)^{2} \) (T^8 + 114T^6 + 2641T^4 + 17824*T^2 + 30976)^2
$47$	\( T^{16} - 33 T^{14} + \cdots + 65536 \) T^16 - 33T^14 + 473T^12 - 2961T^10 + 9025T^8 - 16176T^6 + 132608T^4 + 49152*T^2 + 65536
$53$	\( T^{16} + \cdots + 4294967296 \) T^16 - 167T^14 + 11158T^12 + 115916T^10 + 59314305T^8 + 955607296T^6 + 55624204288T^4 - 25031606272*T^2 + 4294967296
$59$	\( (T^{8} + 10 T^{7} + \cdots + 102400)^{2} \) (T^8 + 10T^7 + 260T^6 + 1320T^5 + 17360T^4 + 105600T^3 + 320000T^2 + 281600*T + 102400)^2
$61$	\( (T^{8} - 6 T^{7} + \cdots + 2920681)^{2} \) (T^8 - 6T^7 - 18T^6 + 332T^5 + 9180T^4 - 56572T^3 + 570837T^2 - 984384*T + 2920681)^2
$67$	\( T^{16} + \cdots + 794123370496 \) T^16 - 353T^14 + 51753T^12 - 1334641T^10 + 462689105T^8 - 12950447296T^6 + 140360683008T^4 + 149382909952*T^2 + 794123370496
$71$	\( (T^{8} + 9 T^{7} + \cdots + 4096)^{2} \) (T^8 + 9T^7 + 217T^6 + 297T^5 + 5225T^4 - 21432T^3 + 32512T^2 + 19456*T + 4096)^2
$73$	\( T^{16} + \cdots + 1380756603136 \) T^16 - 352T^14 + 47318T^12 + 167361T^10 + 43986725T^8 - 317475324T^6 + 7889246048T^4 - 131455864832*T^2 + 1380756603136
$79$	\( (T^{8} - 10 T^{7} + \cdots + 102400)^{2} \) (T^8 - 10T^7 + 120T^6 - 320T^5 - 240T^4 + 3200T^3 + 25600T^2 - 25600*T + 102400)^2
$83$	\( T^{16} + \cdots + 180227832610816 \) T^16 - 77T^14 + 19833T^12 - 3107189T^10 + 283380705T^8 - 6695212304T^6 + 426901737728T^4 - 12797040463872*T^2 + 180227832610816
$89$	\( (T^{8} - 15 T^{7} + \cdots + 9610000)^{2} \) (T^8 - 15T^7 + 70T^6 + 750T^5 + 8525T^4 - 30750T^3 + 1025500T^2 - 2015000*T + 9610000)^2
$97$	\( (T^{8} + 76 T^{6} + 15126 T^{4} + \cdots + 1)^{2} \) (T^8 + 76T^6 + 15126T^4 - 199*T^2 + 1)^2
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\(n\)	\(127\)
\(\chi(n)\)	\(-\beta_{3}\)