LMFDB - Newform orbit 363.3.h.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,3,Mod(245,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 8]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("363.245");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	363.h (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:	$9.89103359628$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	16.0.23612624896000000000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 5 x^{14} + 20 x^{13} + 19 x^{12} + 88 x^{11} - 497 x^{10} + 10 x^{9} + 3711 x^{8} + \cdots + 16 $ x^16 - 2x^15 - 5x^14 + 20x^13 + 19x^12 + 88x^11 - 497x^10 + 10x^9 + 3711x^8 - 3712x^7 + 2758x^6 - 1816x^5 + 1084x^4 - 464x^3 + 184x^2 - 64*x + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
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_{11} q^{2} + ( - \beta_{13} - \beta_{6}) q^{3} + 3 \beta_{9} q^{4} - \beta_{8} q^{5} + (\beta_{7} - \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{12} + \cdots + \beta_1) q^{7}+ \cdots + ( - 2 \beta_{14} - 7 \beta_{9} + \cdots + 7) q^{9}+O(q^{10}) $ q + b11 * q^2 + (-b13 - b6) * q^3 + 3b9 q^4 - b8 * q^5 + (b7 - b5) * q^6 + (-b15 + b12 - b5 + b1) * q^7 - b10 * q^8 + (-2b14 - 7b9 + 7b6 - 7b4 + 7) * q^9	$ q + \beta_{11} q^{2} + ( - \beta_{13} - \beta_{6}) q^{3} + 3 \beta_{9} q^{4} - \beta_{8} q^{5} + (\beta_{7} - \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{12} + \cdots + \beta_1) q^{7}+ \cdots - 7 \beta_{2} q^{98}+O(q^{100}) $ q + b11 * q^2 + (-b13 - b6) * q^3 + 3b9 q^4 - b8 * q^5 + (b7 - b5) * q^6 + (-b15 + b12 - b5 + b1) * q^7 - b10 * q^8 + (-2b14 - 7b9 + 7b6 - 7b4 + 7) * q^9 + b1 * q^10 + (-3b3 + 3) q^12 - 3b15 q^13 - 7b13 q^14 + (b14 - b13 + 8b9 - b8 + b3) q^15 + 19b4 q^16 - 8b7 q^17 + (-2b15 + 2b12 + 7b11 - 7b10 - 7b7 - 2b5 - 7b2 + 2b1) * q^18 + 2b12 q^19 - 3b14 q^20 + (-8b2 + b1) q^21 - 11b3 q^23 + (-b15 - b11) * q^24 + 17b6 q^25 + (21b14 - 21b13 - 21b8 + 21b3) * q^26 + (5b8 - 23b4) * q^27 - 3b5 q^28 + (-8b11 + 8b10 + 8b7 + 8b2) * q^29 + (-b12 + 8b10) q^30 + (30b9 - 30b6 + 30b4 - 30) q^31 + 15b2 q^32 - 56 * q^34 + 8b11 q^35 + (-6b13 + 21b6) * q^36 + 10b9 q^37 + 14b8 q^38 + (24b7 + 3b5) * q^39 + (b15 - b12 + b5 - b1) * q^40 - 16b10 q^41 + (-7b14 - 56b9 + 56b6 - 56b4 + 56) * q^42 - 2b1 q^43 + (-7b3 + 16) q^45 + 11b15 q^46 - 13b13 q^47 + (19b14 - 19b13 - 19b9 - 19b8 + 19b3) q^48 - 7b4 q^49 - 17b7 q^50 + (8b15 - 8b12 + 8b11 - 8b10 - 8b7 + 8b5 - 8b2 - 8b1) * q^51 - 9b12 q^52 - 15b14 q^53 + (-23b2 - 5b1) * q^54 + 7b3 q^56 + (2b15 - 16b11) * q^57 - 56b6 q^58 + (-12b14 + 12b13 + 12b8 - 12b3) * q^59 + (-3b8 - 24b4) * q^60 + 13b5 q^61 + (-30b11 + 30b10 + 30b7 + 30b2) * q^62 + (7b12 + 16b10) * q^63 + (29b9 - 29b6 + 29b4 - 29) q^64 + 24b2 q^65 - 42 * q^67 - 24b11 q^68 + (-11b13 + 88b6) * q^69 + 56b9 q^70 - 23b8 q^71 + (7b7 + 2b5) * q^72 + (10b15 - 10b12 + 10b5 - 10b1) * q^73 + 10b10 q^74 + (17b14 - 17b9 + 17b6 - 17b4 + 17) * q^75 - 6b1 q^76 + (21b3 + 168) q^78 - 3b15 q^79 + 19b13 q^80 + (-28b14 + 28b13 - 17b9 + 28b8 - 28b3) q^81 + 112b4 q^82 - 8b7 q^83 + (-3b15 + 3b12 + 24b11 - 24b10 - 24b7 - 3b5 - 24b2 + 3b1) * q^84 + 8b12 q^85 + 14b14 q^86 + (8b2 + 8b1) * q^87 - 22b3 q^89 + (7b15 + 16b11) * q^90 - 168b6 q^91 + (-33b14 + 33b13 + 33b8 - 33b3) * q^92 + (-30b8 + 30b4) * q^93 - 13b5 q^94 + (16b11 - 16b10 - 16b7 - 16b2) * q^95 + (-15b12 - 15b10) * q^96 + (74b9 - 74b6 + 74b4 - 74) q^97 - 7b2 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{3} + 12 q^{4} + 28 q^{9}+O(q^{10}) $ 16 * q + 4 * q^3 + 12 * q^4 + 28 * q^9	$ 16 q + 4 q^{3} + 12 q^{4} + 28 q^{9} + 48 q^{12} + 32 q^{15} + 76 q^{16} - 68 q^{25} - 92 q^{27} - 120 q^{31} - 896 q^{34} - 84 q^{36} + 40 q^{37} + 224 q^{42} + 256 q^{45} - 76 q^{48} - 28 q^{49} + 224 q^{58} - 96 q^{60} - 116 q^{64} - 672 q^{67} - 352 q^{69} + 224 q^{70} + 68 q^{75} + 2688 q^{78} - 68 q^{81} + 448 q^{82} + 672 q^{91} + 120 q^{93} - 296 q^{97}+O(q^{100}) $ 16 * q + 4 * q^3 + 12 * q^4 + 28 * q^9 + 48 * q^12 + 32 * q^15 + 76 * q^16 - 68 * q^25 - 92 * q^27 - 120 * q^31 - 896 * q^34 - 84 * q^36 + 40 * q^37 + 224 * q^42 + 256 * q^45 - 76 * q^48 - 28 * q^49 + 224 * q^58 - 96 * q^60 - 116 * q^64 - 672 * q^67 - 352 * q^69 + 224 * q^70 + 68 * q^75 + 2688 * q^78 - 68 * q^81 + 448 * q^82 + 672 * q^91 + 120 * q^93 - 296 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2 x^{15} - 5 x^{14} + 20 x^{13} + 19 x^{12} + 88 x^{11} - 497 x^{10} + 10 x^{9} + 3711 x^{8} + \cdots + 16 $ x^16 - 2*x^15 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16 :

$\beta_{1}$	$=$	$ ( -35\nu^{15} - 5454\nu^{10} - 229\nu^{5} + 51646824 ) / 6901598 $ (-35v^15 - 5454v^10 - 229*v^5 + 51646824) / 6901598
$\beta_{2}$	$=$	$ ( 156\nu^{15} + 40874\nu^{10} + 4339168\nu^{5} + 91115 ) / 43681 $ (156v^15 + 40874v^10 + 4339168*v^5 + 91115) / 43681
$\beta_{3}$	$=$	$ ( 73\nu^{15} + 19122\nu^{10} + 2030263\nu^{5} + 42632 ) / 19118 $ (73v^15 + 19122v^10 + 2030263*v^5 + 42632) / 19118
$\beta_{4}$	$=$	$ ( 26318 \nu^{15} + 65795 \nu^{14} - 263180 \nu^{13} - 250021 \nu^{12} + 2290802 \nu^{11} + \cdots - 210544 ) / 27606392 $ (26318v^15 + 65795v^14 - 263180v^13 - 250021v^12 + 2290802v^11 + 6540023v^10 - 131590v^9 - 48833049v^8 + 48846208v^7 + 330169835v^6 + 23896744v^5 - 14264356v^4 + 6105776v^3 - 2421256v^2 + 61967056*v - 210544) / 27606392
$\beta_{5}$	$=$	$ ( 49226 \nu^{15} + 123065 \nu^{14} - 492260 \nu^{13} - 467647 \nu^{12} + 4286694 \nu^{11} + \cdots - 393808 ) / 6901598 $ (49226v^15 + 123065v^14 - 492260v^13 - 467647v^12 + 4286694v^11 + 12232661v^10 - 246130v^9 - 91338843v^8 + 91363456v^7 + 617705661v^6 + 44697208v^5 - 26680492v^4 + 11420432v^3 - 4528792v^2 + 115929412*v - 393808) / 6901598
$\beta_{6}$	$=$	$ ( 229347 \nu^{15} - 254830 \nu^{14} - 1503497 \nu^{13} + 3466982 \nu^{12} + 8180043 \nu^{11} + \cdots - 815456 ) / 13803196 $ (229347v^15 - 254830v^14 - 1503497v^13 + 3466982v^12 + 8180043v^11 + 25075272v^10 - 95077073v^9 - 94541930v^8 + 828160307v^7 - 94287100v^6 + 64930684v^5 - 43423032v^4 + 18959352v^3 + 57125334v^2 + 2854096*v - 815456) / 13803196
$\beta_{7}$	$=$	$ ( - 381690 \nu^{15} - 954225 \nu^{14} + 3816900 \nu^{13} + 3626055 \nu^{12} - 33201382 \nu^{11} + \cdots + 3053520 ) / 27606392 $ (-381690v^15 - 954225v^14 + 3816900v^13 + 3626055v^12 - 33201382v^11 - 94849965v^10 + 1908450v^9 + 708225795v^8 - 708416640v^7 - 4781241081v^6 - 346574520v^5 + 206875980v^4 - 88552080v^3 + 35115480v^2 - 97840880*v + 3053520) / 27606392
$\beta_{8}$	$=$	$ ( 102002 \nu^{15} + 255005 \nu^{14} - 1020020 \nu^{13} - 969019 \nu^{12} + 8873046 \nu^{11} + \cdots - 816016 ) / 6901598 $ (102002v^15 + 255005v^14 - 1020020v^13 - 969019v^12 + 8873046v^11 + 25347497v^10 - 510010v^9 - 189264711v^8 + 189315712v^7 + 1277852729v^6 + 92617816v^5 - 55285084v^4 + 23664464v^3 - 9384184v^2 + 26148788*v - 816016) / 6901598
$\beta_{9}$	$=$	$ ( 1170828 \nu^{15} - 2048949 \nu^{14} - 6436074 \nu^{13} + 21953025 \nu^{12} + 28099872 \nu^{11} + \cdots - 21074904 ) / 27606392 $ (1170828v^15 - 2048949v^14 - 6436074v^13 + 21953025v^12 + 28099872v^11 + 108594297v^10 - 556143300v^9 - 132866291v^8 + 4347869778v^7 - 3259877859v^6 + 2142615240v^5 - 1318937742v^4 + 832294896v^3 - 225969804v^2 + 79616304*v - 21074904) / 27606392
$\beta_{10}$	$=$	$ ( 1606689 \nu^{15} - 1785210 \nu^{14} - 10532739 \nu^{13} + 24273050 \nu^{12} + 57305241 \nu^{11} + \cdots - 5712672 ) / 13803196 $ (1606689v^15 - 1785210v^14 - 10532739v^13 + 24273050v^12 + 57305241v^11 + 175664664v^10 - 666061851v^9 - 662312910v^8 + 5797776305v^7 - 660527700v^6 + 454871508v^5 - 304199784v^4 + 132819624v^3 + 2738v^2 + 19994352*v - 5712672) / 13803196
$\beta_{11}$	$=$	$ ( - 1108238 \nu^{15} + 2241089 \nu^{14} + 5541190 \nu^{13} - 22164760 \nu^{12} - 21056522 \nu^{11} + \cdots + 70927232 ) / 13803196 $ (-1108238v^15 + 2241089v^14 + 5541190v^13 - 22164760v^12 - 21056522v^11 - 97524944v^10 + 557246924v^9 - 11082380v^8 - 4112671218v^7 + 4113779456v^6 - 3056520404v^5 + 2698148523v^4 - 1201329992v^3 + 514222432v^2 - 203915792*v + 70927232) / 13803196
$\beta_{12}$	$=$	$ ( 1716345 \nu^{15} - 1907050 \nu^{14} - 11251595 \nu^{13} + 25945096 \nu^{12} + 61216305 \nu^{11} + \cdots - 6102560 ) / 13803196 $ (1716345v^15 - 1907050v^14 - 11251595v^13 + 25945096v^12 + 61216305v^11 + 187653720v^10 - 711520355v^9 - 707515550v^8 + 6197420205v^7 - 705608500v^6 + 485916340v^5 - 324961320v^4 + 141884520v^3 + 427484520v^2 + 21358960*v - 6102560) / 13803196
$\beta_{13}$	$=$	$ ( - 1717605 \nu^{15} + 1908450 \nu^{14} + 11259855 \nu^{13} - 25949272 \nu^{12} - 61261245 \nu^{11} + \cdots + 6107040 ) / 13803196 $ (-1717605v^15 + 1908450v^14 + 11259855v^13 - 25949272v^12 - 61261245v^11 - 187791480v^10 + 712042695v^9 + 708034950v^8 - 6198074361v^7 + 706126500v^6 - 486273060v^5 + 325199880v^4 - 141988680v^3 - 3528v^2 - 21374640*v + 6107040) / 13803196
$\beta_{14}$	$=$	$ ( 1184757 \nu^{15} - 2395902 \nu^{14} - 5923785 \nu^{13} + 23695140 \nu^{12} + 22510383 \nu^{11} + \cdots - 75824448 ) / 13803196 $ (1184757v^15 - 2395902v^14 - 5923785v^13 + 23695140v^12 + 22510383v^11 + 104258616v^10 - 595732725v^9 + 11847570v^8 + 4396633227v^7 - 4397817984v^6 + 3267559806v^5 - 2884443884v^4 + 1284276588v^3 - 549727248v^2 + 217995288*v - 75824448) / 13803196
$\beta_{15}$	$=$	$ ( - 2763033 \nu^{15} + 5513318 \nu^{14} + 13815165 \nu^{13} - 55260660 \nu^{12} - 52497627 \nu^{11} + \cdots + 176834112 ) / 13803196 $ (-2763033v^15 + 5513318v^14 + 13815165v^13 - 55260660v^12 - 52497627v^11 - 243146904v^10 + 1369878505v^9 - 27630330v^8 - 10253615463v^7 + 10256378496v^6 - 7620445014v^5 + 4663433948v^4 - 2995127772v^3 + 1282047312v^2 - 508398072*v + 176834112) / 13803196

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

$\nu$	$=$	$ ( -\beta_{8} - \beta_{7} + \beta_{4} ) / 2 $ (-b8 - b7 + b4) / 2
$\nu^{2}$	$=$	$ ( \beta_{13} + \beta_{12} + \beta_{10} - 7\beta_{6} ) / 2 $ (b13 + b12 + b10 - 7*b6) / 2
$\nu^{3}$	$=$	$ ( - 3 \beta_{15} + 13 \beta_{14} - 13 \beta_{13} + 3 \beta_{12} + 14 \beta_{11} - 14 \beta_{10} + \cdots + 3 \beta_1 ) / 4 $ (-3b15 + 13b14 - 13b13 + 3b12 + 14b11 - 14b10 - 22b9 - 13b8 - 14b7 - 3b5 + 13b3 - 14b2 + 3*b1) / 4
$\nu^{4}$	$=$	$ ( 6\beta_{15} + 14\beta_{14} + 15\beta_{11} + 45\beta_{9} - 45\beta_{6} + 45\beta_{4} - 45 ) / 2 $ (6b15 + 14b14 + 15b11 + 45b9 - 45b6 + 45b4 - 45) / 2
$\nu^{5}$	$=$	$ ( 73\beta_{3} - 78\beta_{2} + 35\beta _1 - 262 ) / 4 $ (73b3 - 78b2 + 35*b1 - 262) / 4
$\nu^{6}$	$=$	$ ( 145\beta_{8} + 155\beta_{7} - 29\beta_{5} + 217\beta_{4} ) / 2 $ (145b8 + 155b7 - 29b5 + 217b4) / 2
$\nu^{7}$	$=$	$ ( 275\beta_{13} - 329\beta_{12} + 294\beta_{10} + 2462\beta_{6} ) / 4 $ (275b13 - 329b12 + 294b10 + 2462b6) / 4
$\nu^{8}$	$=$	$ ( - 60 \beta_{15} - 1260 \beta_{14} + 1260 \beta_{13} + 60 \beta_{12} - 1347 \beta_{11} + 1347 \beta_{10} + \cdots + 60 \beta_1 ) / 2 $ (-60b15 - 1260b14 + 1260b13 + 60b12 - 1347b11 + 1347b10 - 449b9 + 1260b8 + 1347b7 - 60b5 - 1260b3 + 1347b2 + 60*b1) / 2
$\nu^{9}$	$=$	$ ( -2667\beta_{15} - 391\beta_{14} - 418\beta_{11} - 19958\beta_{9} + 19958\beta_{6} - 19958\beta_{4} + 19958 ) / 4 $ (-2667b15 - 391b14 - 418b11 - 19958b9 + 19958b6 - 19958b4 + 19958) / 4
$\nu^{10}$	$=$	$ ( -9559\beta_{3} + 10219\beta_{2} - 869\beta _1 + 6503 ) / 2 $ (-9559b3 + 10219b2 - 869*b1 + 6503) / 2
$\nu^{11}$	$=$	$ ( -22145\beta_{8} - 23674\beta_{7} + 18909\beta_{5} - 141502\beta_{4} ) / 4 $ (-22145b8 - 23674b7 + 18909b5 - 141502b4) / 4
$\nu^{12}$	$=$	$ ( -62930\beta_{13} + 16182\beta_{12} - 67275\beta_{10} - 121095\beta_{6} ) / 2 $ (-62930b13 + 16182b12 - 67275b10 - 121095b6) / 2
$\nu^{13}$	$=$	$ ( 114023 \beta_{15} + 297299 \beta_{14} - 297299 \beta_{13} - 114023 \beta_{12} + 317826 \beta_{11} + \cdots - 114023 \beta_1 ) / 4 $ (114023b15 + 297299b14 - 297299b13 - 114023b12 + 317826b11 - 317826b10 + 853270b9 - 297299b8 - 317826b7 + 114023b5 + 297299b3 - 317826b2 - 114023*b1) / 4
$\nu^{14}$	$=$	$ ( 182287 \beta_{15} - 338533 \beta_{14} - 361907 \beta_{11} + 1364111 \beta_{9} - 1364111 \beta_{6} + \cdots - 1364111 ) / 2 $ (182287b15 - 338533b14 - 361907b11 + 1364111b9 - 1364111b6 + 1364111b4 - 1364111) / 2
$\nu^{15}$	$=$	$ ( 2978653\beta_{3} - 3184314\beta_{2} - 518153\beta _1 + 3877502 ) / 4 $ (2978653b3 - 3184314b2 - 518153*b1 + 3877502) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$-1$	$-\beta_{9}$

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} $

245.1

2.01333	−	1.92046i
0.350821	+	0.367787i
0.458196	+	0.219999i
−1.20431	+	2.50824i
−0.241376	−	0.447303i
2.44862	−	1.32134i
−2.75764	+	0.370279i
−0.0676410	−	0.503753i
−0.241376	+	0.447303i
2.44862	+	1.32134i
−2.75764	−	0.370279i
−0.0676410	+	0.503753i
2.01333	+	1.92046i
0.350821	−	0.367787i
0.458196	−	0.219999i
−1.20431	−	2.50824i

−1.55513

−

2.14046i

−2.99901

−

0.0770245i

−0.927051

2.85317i

1.66251

−

2.28825i

4.49900

6.53904i

2.31247

−

7.11706i

−2.51626

0.817582i

8.98813

0.461994i

−7.48331

245.2

−1.55513

−

2.14046i

2.38098

−

1.82509i

−0.927051

2.85317i

−1.66251

2.28825i

−7.60926

−

2.25812i

−2.31247

7.11706i

−2.51626

0.817582i

2.33810

−

8.69099i

7.48331

245.3

1.55513

2.14046i

−2.99901

−

0.0770245i

−0.927051

2.85317i

1.66251

−

2.28825i

−4.49900

−

6.53904i

−2.31247

7.11706i

2.51626

−

0.817582i

8.98813

0.461994i

7.48331

245.4

1.55513

2.14046i

2.38098

−

1.82509i

−0.927051

2.85317i

−1.66251

2.28825i

7.60926

2.25812i

2.31247

−

7.11706i

2.51626

−

0.817582i

2.33810

−

8.69099i

−7.48331

251.1

−2.51626

−

0.817582i

−0.853491

−

2.87603i

2.42705

1.76336i

−2.68999

0.874032i

−0.203788

7.93464i

6.05413

4.39858i

1.55513

2.14046i

−7.54311

4.90933i

7.48331

251.2

−2.51626

−

0.817582i

2.47152

1.70046i

2.42705

1.76336i

2.68999

−

0.874032i

−4.82873

−

6.29947i

−6.05413

−

4.39858i

1.55513

2.14046i

3.21687

8.40546i

−7.48331

251.3

2.51626

0.817582i

−0.853491

−

2.87603i

2.42705

1.76336i

−2.68999

0.874032i

0.203788

−

7.93464i

−6.05413

−

4.39858i

−1.55513

−

2.14046i

−7.54311

4.90933i

−7.48331

251.4

2.51626

0.817582i

2.47152

1.70046i

2.42705

1.76336i

2.68999

−

0.874032i

4.82873

6.29947i

6.05413

4.39858i

−1.55513

−

2.14046i

3.21687

8.40546i

7.48331

269.1

−2.51626

0.817582i

−0.853491

2.87603i

2.42705

−

1.76336i

−2.68999

−

0.874032i

−0.203788

−

7.93464i

6.05413

−

4.39858i

1.55513

−

2.14046i

−7.54311

−

4.90933i

7.48331

269.2

−2.51626

0.817582i

2.47152

−

1.70046i

2.42705

−

1.76336i

2.68999

0.874032i

−4.82873

6.29947i

−6.05413

4.39858i

1.55513

−

2.14046i

3.21687

−

8.40546i

−7.48331

269.3

2.51626

−

0.817582i

−0.853491

2.87603i

2.42705

−

1.76336i

−2.68999

−

0.874032i

0.203788

7.93464i

−6.05413

4.39858i

−1.55513

2.14046i

−7.54311

−

4.90933i

−7.48331

269.4

2.51626

−

0.817582i

2.47152

−

1.70046i

2.42705

−

1.76336i

2.68999

0.874032i

4.82873

−

6.29947i

6.05413

−

4.39858i

−1.55513

2.14046i

3.21687

−

8.40546i

7.48331

323.1

−1.55513

2.14046i

−2.99901

0.0770245i

−0.927051

−

2.85317i

1.66251

2.28825i

4.49900

−

6.53904i

2.31247

7.11706i

−2.51626

−

0.817582i

8.98813

−

0.461994i

−7.48331

323.2

−1.55513

2.14046i

2.38098

1.82509i

−0.927051

−

2.85317i

−1.66251

−

2.28825i

−7.60926

2.25812i

−2.31247

−

7.11706i

−2.51626

−

0.817582i

2.33810

8.69099i

7.48331

323.3

1.55513

−

2.14046i

−2.99901

0.0770245i

−0.927051

−

2.85317i

1.66251

2.28825i

−4.49900

6.53904i

−2.31247

−

7.11706i

2.51626

0.817582i

8.98813

−

0.461994i

7.48331

323.4

1.55513

−

2.14046i

2.38098

1.82509i

−0.927051

−

2.85317i

−1.66251

−

2.28825i

7.60926

−

2.25812i

2.31247

7.11706i

2.51626

0.817582i

2.33810

8.69099i

−7.48331

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
33.d	even	2	1	inner
33.f	even	10	3	inner
33.h	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	363.3.h.k		16
3.b	odd	2	1	inner	363.3.h.k		16
11.b	odd	2	1	inner	363.3.h.k		16
11.c	even	5	1		363.3.b.i	✓	4
11.c	even	5	3	inner	363.3.h.k		16
11.d	odd	10	1		363.3.b.i	✓	4
11.d	odd	10	3	inner	363.3.h.k		16
33.d	even	2	1	inner	363.3.h.k		16
33.f	even	10	1		363.3.b.i	✓	4
33.f	even	10	3	inner	363.3.h.k		16
33.h	odd	10	1		363.3.b.i	✓	4
33.h	odd	10	3	inner	363.3.h.k		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
363.3.b.i	✓	4	11.c	even	5	1
363.3.b.i	✓	4	11.d	odd	10	1
363.3.b.i	✓	4	33.f	even	10	1
363.3.b.i	✓	4	33.h	odd	10	1
363.3.h.k		16	1.a	even	1	1	trivial
363.3.h.k		16	3.b	odd	2	1	inner
363.3.h.k		16	11.b	odd	2	1	inner
363.3.h.k		16	11.c	even	5	3	inner
363.3.h.k		16	11.d	odd	10	3	inner
363.3.h.k		16	33.d	even	2	1	inner
363.3.h.k		16	33.f	even	10	3	inner
363.3.h.k		16	33.h	odd	10	3	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(363, [\chi])$:

$ T_{2}^{8} - 7T_{2}^{6} + 49T_{2}^{4} - 343T_{2}^{2} + 2401 $

$ T_{5}^{8} - 8T_{5}^{6} + 64T_{5}^{4} - 512T_{5}^{2} + 4096 $

$ T_{7}^{8} + 56T_{7}^{6} + 3136T_{7}^{4} + 175616T_{7}^{2} + 9834496 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - 7 T^{6} + \cdots + 2401)^{2} $ (T^8 - 7T^6 + 49T^4 - 343*T^2 + 2401)^2
$3$	$ (T^{8} - 2 T^{7} + \cdots + 6561)^{2} $ (T^8 - 2T^7 - 5T^6 + 28T^5 - 11T^4 + 252T^3 - 405T^2 - 1458*T + 6561)^2
$5$	$ (T^{8} - 8 T^{6} + \cdots + 4096)^{2} $ (T^8 - 8T^6 + 64T^4 - 512*T^2 + 4096)^2
$7$	$ (T^{8} + 56 T^{6} + \cdots + 9834496)^{2} $ (T^8 + 56T^6 + 3136T^4 + 175616*T^2 + 9834496)^2
$11$	$ T^{16} $ T^16
$13$	$ (T^{8} + 504 T^{6} + \cdots + 64524128256)^{2} $ (T^8 + 504T^6 + 254016T^4 + 128024064*T^2 + 64524128256)^2
$17$	$ (T^{8} - 448 T^{6} + \cdots + 40282095616)^{2} $ (T^8 - 448T^6 + 200704T^4 - 89915392*T^2 + 40282095616)^2
$19$	$ (T^{8} + 224 T^{6} + \cdots + 2517630976)^{2} $ (T^8 + 224T^6 + 50176T^4 + 11239424*T^2 + 2517630976)^2
$23$	$ (T^{2} + 968)^{8} $ (T^2 + 968)^8
$29$	$ (T^{8} - 448 T^{6} + \cdots + 40282095616)^{2} $ (T^8 - 448T^6 + 200704T^4 - 89915392*T^2 + 40282095616)^2
$31$	$ (T^{4} + 30 T^{3} + \cdots + 810000)^{4} $ (T^4 + 30T^3 + 900T^2 + 27000*T + 810000)^4
$37$	$ (T^{4} - 10 T^{3} + \cdots + 10000)^{4} $ (T^4 - 10T^3 + 100T^2 - 1000*T + 10000)^4
$41$	$ (T^{8} + \cdots + 10312216477696)^{2} $ (T^8 - 1792T^6 + 3211264T^4 - 5754585088*T^2 + 10312216477696)^2
$43$	$ (T^{2} - 224)^{8} $ (T^2 - 224)^8
$47$	$ (T^{8} + \cdots + 3341233033216)^{2} $ (T^8 - 1352T^6 + 1827904T^4 - 2471326208*T^2 + 3341233033216)^2
$53$	$ (T^{8} + \cdots + 10497600000000)^{2} $ (T^8 - 1800T^6 + 3240000T^4 - 5832000000*T^2 + 10497600000000)^2
$59$	$ (T^{8} + \cdots + 1761205026816)^{2} $ (T^8 - 1152T^6 + 1327104T^4 - 1528823808*T^2 + 1761205026816)^2
$61$	$ (T^{8} + \cdots + 80\!\cdots\!16)^{2} $ (T^8 + 9464T^6 + 89567296T^4 + 847664889344*T^2 + 8022300512751616)^2
$67$	$ (T + 42)^{16} $ (T + 42)^16
$71$	$ (T^{8} + \cdots + 320761795710976)^{2} $ (T^8 - 4232T^6 + 17909824T^4 - 75794375168*T^2 + 320761795710976)^2
$73$	$ (T^{8} + \cdots + 983449600000000)^{2} $ (T^8 + 5600T^6 + 31360000T^4 + 175616000000*T^2 + 983449600000000)^2
$79$	$ (T^{8} + 504 T^{6} + \cdots + 64524128256)^{2} $ (T^8 + 504T^6 + 254016T^4 + 128024064*T^2 + 64524128256)^2
$83$	$ (T^{8} - 448 T^{6} + \cdots + 40282095616)^{2} $ (T^8 - 448T^6 + 200704T^4 - 89915392*T^2 + 40282095616)^2
$89$	$ (T^{2} + 3872)^{8} $ (T^2 + 3872)^8
$97$	$ (T^{4} + 74 T^{3} + \cdots + 29986576)^{4} $ (T^4 + 74T^3 + 5476T^2 + 405224*T + 29986576)^4
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.h (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{11} q^{2} + ( - \beta_{13} - \beta_{6}) q^{3} + 3 \beta_{9} q^{4} - \beta_{8} q^{5} + (\beta_{7} - \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{12} + \cdots + \beta_1) q^{7}+ \cdots + ( - 2 \beta_{14} - 7 \beta_{9} + \cdots + 7) q^{9}+O(q^{10}) \) q + b11 * q^2 + (-b13 - b6) * q^3 + 3b9 q^4 - b8 * q^5 + (b7 - b5) * q^6 + (-b15 + b12 - b5 + b1) * q^7 - b10 * q^8 + (-2b14 - 7b9 + 7b6 - 7b4 + 7) * q^9	\( q + \beta_{11} q^{2} + ( - \beta_{13} - \beta_{6}) q^{3} + 3 \beta_{9} q^{4} - \beta_{8} q^{5} + (\beta_{7} - \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{12} + \cdots + \beta_1) q^{7}+ \cdots - 7 \beta_{2} q^{98}+O(q^{100}) \) q + b11 * q^2 + (-b13 - b6) * q^3 + 3b9 q^4 - b8 * q^5 + (b7 - b5) * q^6 + (-b15 + b12 - b5 + b1) * q^7 - b10 * q^8 + (-2b14 - 7b9 + 7b6 - 7b4 + 7) * q^9 + b1 * q^10 + (-3b3 + 3) q^12 - 3b15 q^13 - 7b13 q^14 + (b14 - b13 + 8b9 - b8 + b3) q^15 + 19b4 q^16 - 8b7 q^17 + (-2b15 + 2b12 + 7b11 - 7b10 - 7b7 - 2b5 - 7b2 + 2b1) * q^18 + 2b12 q^19 - 3b14 q^20 + (-8b2 + b1) q^21 - 11b3 q^23 + (-b15 - b11) * q^24 + 17b6 q^25 + (21b14 - 21b13 - 21b8 + 21b3) * q^26 + (5b8 - 23b4) * q^27 - 3b5 q^28 + (-8b11 + 8b10 + 8b7 + 8b2) * q^29 + (-b12 + 8b10) q^30 + (30b9 - 30b6 + 30b4 - 30) q^31 + 15b2 q^32 - 56 * q^34 + 8b11 q^35 + (-6b13 + 21b6) * q^36 + 10b9 q^37 + 14b8 q^38 + (24b7 + 3b5) * q^39 + (b15 - b12 + b5 - b1) * q^40 - 16b10 q^41 + (-7b14 - 56b9 + 56b6 - 56b4 + 56) * q^42 - 2b1 q^43 + (-7b3 + 16) q^45 + 11b15 q^46 - 13b13 q^47 + (19b14 - 19b13 - 19b9 - 19b8 + 19b3) q^48 - 7b4 q^49 - 17b7 q^50 + (8b15 - 8b12 + 8b11 - 8b10 - 8b7 + 8b5 - 8b2 - 8b1) * q^51 - 9b12 q^52 - 15b14 q^53 + (-23b2 - 5b1) * q^54 + 7b3 q^56 + (2b15 - 16b11) * q^57 - 56b6 q^58 + (-12b14 + 12b13 + 12b8 - 12b3) * q^59 + (-3b8 - 24b4) * q^60 + 13b5 q^61 + (-30b11 + 30b10 + 30b7 + 30b2) * q^62 + (7b12 + 16b10) * q^63 + (29b9 - 29b6 + 29b4 - 29) q^64 + 24b2 q^65 - 42 * q^67 - 24b11 q^68 + (-11b13 + 88b6) * q^69 + 56b9 q^70 - 23b8 q^71 + (7b7 + 2b5) * q^72 + (10b15 - 10b12 + 10b5 - 10b1) * q^73 + 10b10 q^74 + (17b14 - 17b9 + 17b6 - 17b4 + 17) * q^75 - 6b1 q^76 + (21b3 + 168) q^78 - 3b15 q^79 + 19b13 q^80 + (-28b14 + 28b13 - 17b9 + 28b8 - 28b3) q^81 + 112b4 q^82 - 8b7 q^83 + (-3b15 + 3b12 + 24b11 - 24b10 - 24b7 - 3b5 - 24b2 + 3b1) * q^84 + 8b12 q^85 + 14b14 q^86 + (8b2 + 8b1) * q^87 - 22b3 q^89 + (7b15 + 16b11) * q^90 - 168b6 q^91 + (-33b14 + 33b13 + 33b8 - 33b3) * q^92 + (-30b8 + 30b4) * q^93 - 13b5 q^94 + (16b11 - 16b10 - 16b7 - 16b2) * q^95 + (-15b12 - 15b10) * q^96 + (74b9 - 74b6 + 74b4 - 74) q^97 - 7b2 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{3} + 12 q^{4} + 28 q^{9}+O(q^{10}) \) 16 * q + 4 * q^3 + 12 * q^4 + 28 * q^9	\( 16 q + 4 q^{3} + 12 q^{4} + 28 q^{9} + 48 q^{12} + 32 q^{15} + 76 q^{16} - 68 q^{25} - 92 q^{27} - 120 q^{31} - 896 q^{34} - 84 q^{36} + 40 q^{37} + 224 q^{42} + 256 q^{45} - 76 q^{48} - 28 q^{49} + 224 q^{58} - 96 q^{60} - 116 q^{64} - 672 q^{67} - 352 q^{69} + 224 q^{70} + 68 q^{75} + 2688 q^{78} - 68 q^{81} + 448 q^{82} + 672 q^{91} + 120 q^{93} - 296 q^{97}+O(q^{100}) \) 16 * q + 4 * q^3 + 12 * q^4 + 28 * q^9 + 48 * q^12 + 32 * q^15 + 76 * q^16 - 68 * q^25 - 92 * q^27 - 120 * q^31 - 896 * q^34 - 84 * q^36 + 40 * q^37 + 224 * q^42 + 256 * q^45 - 76 * q^48 - 28 * q^49 + 224 * q^58 - 96 * q^60 - 116 * q^64 - 672 * q^67 - 352 * q^69 + 224 * q^70 + 68 * q^75 + 2688 * q^78 - 68 * q^81 + 448 * q^82 + 672 * q^91 + 120 * q^93 - 296 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	363.3.h.k

Level	$363$
Weight	$3$
Character orbit	363.h
Analytic conductor	$9.891$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(9.89103359628\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	16.0.23612624896000000000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 2 x^{15} - 5 x^{14} + 20 x^{13} + 19 x^{12} + 88 x^{11} - 497 x^{10} + 10 x^{9} + 3711 x^{8} + \cdots + 16 \) x^16 - 2x^15 - 5x^14 + 20x^13 + 19x^12 + 88x^11 - 497x^10 + 10x^9 + 3711x^8 - 3712x^7 + 2758x^6 - 1816x^5 + 1084x^4 - 464x^3 + 184x^2 - 64*x + 16
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(\beta_{1}\)	\(=\)	\( ( -35\nu^{15} - 5454\nu^{10} - 229\nu^{5} + 51646824 ) / 6901598 \) (-35v^15 - 5454v^10 - 229*v^5 + 51646824) / 6901598
\(\beta_{2}\)	\(=\)	\( ( 156\nu^{15} + 40874\nu^{10} + 4339168\nu^{5} + 91115 ) / 43681 \) (156v^15 + 40874v^10 + 4339168*v^5 + 91115) / 43681
\(\beta_{3}\)	\(=\)	\( ( 73\nu^{15} + 19122\nu^{10} + 2030263\nu^{5} + 42632 ) / 19118 \) (73v^15 + 19122v^10 + 2030263*v^5 + 42632) / 19118
\(\beta_{4}\)	\(=\)	\( ( 26318 \nu^{15} + 65795 \nu^{14} - 263180 \nu^{13} - 250021 \nu^{12} + 2290802 \nu^{11} + \cdots - 210544 ) / 27606392 \) (26318v^15 + 65795v^14 - 263180v^13 - 250021v^12 + 2290802v^11 + 6540023v^10 - 131590v^9 - 48833049v^8 + 48846208v^7 + 330169835v^6 + 23896744v^5 - 14264356v^4 + 6105776v^3 - 2421256v^2 + 61967056*v - 210544) / 27606392
\(\beta_{5}\)	\(=\)	\( ( 49226 \nu^{15} + 123065 \nu^{14} - 492260 \nu^{13} - 467647 \nu^{12} + 4286694 \nu^{11} + \cdots - 393808 ) / 6901598 \) (49226v^15 + 123065v^14 - 492260v^13 - 467647v^12 + 4286694v^11 + 12232661v^10 - 246130v^9 - 91338843v^8 + 91363456v^7 + 617705661v^6 + 44697208v^5 - 26680492v^4 + 11420432v^3 - 4528792v^2 + 115929412*v - 393808) / 6901598
\(\beta_{6}\)	\(=\)	\( ( 229347 \nu^{15} - 254830 \nu^{14} - 1503497 \nu^{13} + 3466982 \nu^{12} + 8180043 \nu^{11} + \cdots - 815456 ) / 13803196 \) (229347v^15 - 254830v^14 - 1503497v^13 + 3466982v^12 + 8180043v^11 + 25075272v^10 - 95077073v^9 - 94541930v^8 + 828160307v^7 - 94287100v^6 + 64930684v^5 - 43423032v^4 + 18959352v^3 + 57125334v^2 + 2854096*v - 815456) / 13803196
\(\beta_{7}\)	\(=\)	\( ( - 381690 \nu^{15} - 954225 \nu^{14} + 3816900 \nu^{13} + 3626055 \nu^{12} - 33201382 \nu^{11} + \cdots + 3053520 ) / 27606392 \) (-381690v^15 - 954225v^14 + 3816900v^13 + 3626055v^12 - 33201382v^11 - 94849965v^10 + 1908450v^9 + 708225795v^8 - 708416640v^7 - 4781241081v^6 - 346574520v^5 + 206875980v^4 - 88552080v^3 + 35115480v^2 - 97840880*v + 3053520) / 27606392
\(\beta_{8}\)	\(=\)	\( ( 102002 \nu^{15} + 255005 \nu^{14} - 1020020 \nu^{13} - 969019 \nu^{12} + 8873046 \nu^{11} + \cdots - 816016 ) / 6901598 \) (102002v^15 + 255005v^14 - 1020020v^13 - 969019v^12 + 8873046v^11 + 25347497v^10 - 510010v^9 - 189264711v^8 + 189315712v^7 + 1277852729v^6 + 92617816v^5 - 55285084v^4 + 23664464v^3 - 9384184v^2 + 26148788*v - 816016) / 6901598
\(\beta_{9}\)	\(=\)	\( ( 1170828 \nu^{15} - 2048949 \nu^{14} - 6436074 \nu^{13} + 21953025 \nu^{12} + 28099872 \nu^{11} + \cdots - 21074904 ) / 27606392 \) (1170828v^15 - 2048949v^14 - 6436074v^13 + 21953025v^12 + 28099872v^11 + 108594297v^10 - 556143300v^9 - 132866291v^8 + 4347869778v^7 - 3259877859v^6 + 2142615240v^5 - 1318937742v^4 + 832294896v^3 - 225969804v^2 + 79616304*v - 21074904) / 27606392
\(\beta_{10}\)	\(=\)	\( ( 1606689 \nu^{15} - 1785210 \nu^{14} - 10532739 \nu^{13} + 24273050 \nu^{12} + 57305241 \nu^{11} + \cdots - 5712672 ) / 13803196 \) (1606689v^15 - 1785210v^14 - 10532739v^13 + 24273050v^12 + 57305241v^11 + 175664664v^10 - 666061851v^9 - 662312910v^8 + 5797776305v^7 - 660527700v^6 + 454871508v^5 - 304199784v^4 + 132819624v^3 + 2738v^2 + 19994352*v - 5712672) / 13803196
\(\beta_{11}\)	\(=\)	\( ( - 1108238 \nu^{15} + 2241089 \nu^{14} + 5541190 \nu^{13} - 22164760 \nu^{12} - 21056522 \nu^{11} + \cdots + 70927232 ) / 13803196 \) (-1108238v^15 + 2241089v^14 + 5541190v^13 - 22164760v^12 - 21056522v^11 - 97524944v^10 + 557246924v^9 - 11082380v^8 - 4112671218v^7 + 4113779456v^6 - 3056520404v^5 + 2698148523v^4 - 1201329992v^3 + 514222432v^2 - 203915792*v + 70927232) / 13803196
\(\beta_{12}\)	\(=\)	\( ( 1716345 \nu^{15} - 1907050 \nu^{14} - 11251595 \nu^{13} + 25945096 \nu^{12} + 61216305 \nu^{11} + \cdots - 6102560 ) / 13803196 \) (1716345v^15 - 1907050v^14 - 11251595v^13 + 25945096v^12 + 61216305v^11 + 187653720v^10 - 711520355v^9 - 707515550v^8 + 6197420205v^7 - 705608500v^6 + 485916340v^5 - 324961320v^4 + 141884520v^3 + 427484520v^2 + 21358960*v - 6102560) / 13803196
\(\beta_{13}\)	\(=\)	\( ( - 1717605 \nu^{15} + 1908450 \nu^{14} + 11259855 \nu^{13} - 25949272 \nu^{12} - 61261245 \nu^{11} + \cdots + 6107040 ) / 13803196 \) (-1717605v^15 + 1908450v^14 + 11259855v^13 - 25949272v^12 - 61261245v^11 - 187791480v^10 + 712042695v^9 + 708034950v^8 - 6198074361v^7 + 706126500v^6 - 486273060v^5 + 325199880v^4 - 141988680v^3 - 3528v^2 - 21374640*v + 6107040) / 13803196
\(\beta_{14}\)	\(=\)	\( ( 1184757 \nu^{15} - 2395902 \nu^{14} - 5923785 \nu^{13} + 23695140 \nu^{12} + 22510383 \nu^{11} + \cdots - 75824448 ) / 13803196 \) (1184757v^15 - 2395902v^14 - 5923785v^13 + 23695140v^12 + 22510383v^11 + 104258616v^10 - 595732725v^9 + 11847570v^8 + 4396633227v^7 - 4397817984v^6 + 3267559806v^5 - 2884443884v^4 + 1284276588v^3 - 549727248v^2 + 217995288*v - 75824448) / 13803196
\(\beta_{15}\)	\(=\)	\( ( - 2763033 \nu^{15} + 5513318 \nu^{14} + 13815165 \nu^{13} - 55260660 \nu^{12} - 52497627 \nu^{11} + \cdots + 176834112 ) / 13803196 \) (-2763033v^15 + 5513318v^14 + 13815165v^13 - 55260660v^12 - 52497627v^11 - 243146904v^10 + 1369878505v^9 - 27630330v^8 - 10253615463v^7 + 10256378496v^6 - 7620445014v^5 + 4663433948v^4 - 2995127772v^3 + 1282047312v^2 - 508398072*v + 176834112) / 13803196

\(\nu\)	\(=\)	\( ( -\beta_{8} - \beta_{7} + \beta_{4} ) / 2 \) (-b8 - b7 + b4) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{13} + \beta_{12} + \beta_{10} - 7\beta_{6} ) / 2 \) (b13 + b12 + b10 - 7*b6) / 2
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{15} + 13 \beta_{14} - 13 \beta_{13} + 3 \beta_{12} + 14 \beta_{11} - 14 \beta_{10} + \cdots + 3 \beta_1 ) / 4 \) (-3b15 + 13b14 - 13b13 + 3b12 + 14b11 - 14b10 - 22b9 - 13b8 - 14b7 - 3b5 + 13b3 - 14b2 + 3*b1) / 4
\(\nu^{4}\)	\(=\)	\( ( 6\beta_{15} + 14\beta_{14} + 15\beta_{11} + 45\beta_{9} - 45\beta_{6} + 45\beta_{4} - 45 ) / 2 \) (6b15 + 14b14 + 15b11 + 45b9 - 45b6 + 45b4 - 45) / 2
\(\nu^{5}\)	\(=\)	\( ( 73\beta_{3} - 78\beta_{2} + 35\beta _1 - 262 ) / 4 \) (73b3 - 78b2 + 35*b1 - 262) / 4
\(\nu^{6}\)	\(=\)	\( ( 145\beta_{8} + 155\beta_{7} - 29\beta_{5} + 217\beta_{4} ) / 2 \) (145b8 + 155b7 - 29b5 + 217b4) / 2
\(\nu^{7}\)	\(=\)	\( ( 275\beta_{13} - 329\beta_{12} + 294\beta_{10} + 2462\beta_{6} ) / 4 \) (275b13 - 329b12 + 294b10 + 2462b6) / 4
\(\nu^{8}\)	\(=\)	\( ( - 60 \beta_{15} - 1260 \beta_{14} + 1260 \beta_{13} + 60 \beta_{12} - 1347 \beta_{11} + 1347 \beta_{10} + \cdots + 60 \beta_1 ) / 2 \) (-60b15 - 1260b14 + 1260b13 + 60b12 - 1347b11 + 1347b10 - 449b9 + 1260b8 + 1347b7 - 60b5 - 1260b3 + 1347b2 + 60*b1) / 2
\(\nu^{9}\)	\(=\)	\( ( -2667\beta_{15} - 391\beta_{14} - 418\beta_{11} - 19958\beta_{9} + 19958\beta_{6} - 19958\beta_{4} + 19958 ) / 4 \) (-2667b15 - 391b14 - 418b11 - 19958b9 + 19958b6 - 19958b4 + 19958) / 4
\(\nu^{10}\)	\(=\)	\( ( -9559\beta_{3} + 10219\beta_{2} - 869\beta _1 + 6503 ) / 2 \) (-9559b3 + 10219b2 - 869*b1 + 6503) / 2
\(\nu^{11}\)	\(=\)	\( ( -22145\beta_{8} - 23674\beta_{7} + 18909\beta_{5} - 141502\beta_{4} ) / 4 \) (-22145b8 - 23674b7 + 18909b5 - 141502b4) / 4
\(\nu^{12}\)	\(=\)	\( ( -62930\beta_{13} + 16182\beta_{12} - 67275\beta_{10} - 121095\beta_{6} ) / 2 \) (-62930b13 + 16182b12 - 67275b10 - 121095b6) / 2
\(\nu^{13}\)	\(=\)	\( ( 114023 \beta_{15} + 297299 \beta_{14} - 297299 \beta_{13} - 114023 \beta_{12} + 317826 \beta_{11} + \cdots - 114023 \beta_1 ) / 4 \) (114023b15 + 297299b14 - 297299b13 - 114023b12 + 317826b11 - 317826b10 + 853270b9 - 297299b8 - 317826b7 + 114023b5 + 297299b3 - 317826b2 - 114023*b1) / 4
\(\nu^{14}\)	\(=\)	\( ( 182287 \beta_{15} - 338533 \beta_{14} - 361907 \beta_{11} + 1364111 \beta_{9} - 1364111 \beta_{6} + \cdots - 1364111 ) / 2 \) (182287b15 - 338533b14 - 361907b11 + 1364111b9 - 1364111b6 + 1364111b4 - 1364111) / 2
\(\nu^{15}\)	\(=\)	\( ( 2978653\beta_{3} - 3184314\beta_{2} - 518153\beta _1 + 3877502 ) / 4 \) (2978653b3 - 3184314b2 - 518153*b1 + 3877502) / 4

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 245.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - 7 T^{6} + \cdots + 2401)^{2} \) (T^8 - 7T^6 + 49T^4 - 343*T^2 + 2401)^2
$3$	\( (T^{8} - 2 T^{7} + \cdots + 6561)^{2} \) (T^8 - 2T^7 - 5T^6 + 28T^5 - 11T^4 + 252T^3 - 405T^2 - 1458*T + 6561)^2
$5$	\( (T^{8} - 8 T^{6} + \cdots + 4096)^{2} \) (T^8 - 8T^6 + 64T^4 - 512*T^2 + 4096)^2
$7$	\( (T^{8} + 56 T^{6} + \cdots + 9834496)^{2} \) (T^8 + 56T^6 + 3136T^4 + 175616*T^2 + 9834496)^2
$11$	\( T^{16} \) T^16
$13$	\( (T^{8} + 504 T^{6} + \cdots + 64524128256)^{2} \) (T^8 + 504T^6 + 254016T^4 + 128024064*T^2 + 64524128256)^2
$17$	\( (T^{8} - 448 T^{6} + \cdots + 40282095616)^{2} \) (T^8 - 448T^6 + 200704T^4 - 89915392*T^2 + 40282095616)^2
$19$	\( (T^{8} + 224 T^{6} + \cdots + 2517630976)^{2} \) (T^8 + 224T^6 + 50176T^4 + 11239424*T^2 + 2517630976)^2
$23$	\( (T^{2} + 968)^{8} \) (T^2 + 968)^8
$29$	\( (T^{8} - 448 T^{6} + \cdots + 40282095616)^{2} \) (T^8 - 448T^6 + 200704T^4 - 89915392*T^2 + 40282095616)^2
$31$	\( (T^{4} + 30 T^{3} + \cdots + 810000)^{4} \) (T^4 + 30T^3 + 900T^2 + 27000*T + 810000)^4
$37$	\( (T^{4} - 10 T^{3} + \cdots + 10000)^{4} \) (T^4 - 10T^3 + 100T^2 - 1000*T + 10000)^4
$41$	\( (T^{8} + \cdots + 10312216477696)^{2} \) (T^8 - 1792T^6 + 3211264T^4 - 5754585088*T^2 + 10312216477696)^2
$43$	\( (T^{2} - 224)^{8} \) (T^2 - 224)^8
$47$	\( (T^{8} + \cdots + 3341233033216)^{2} \) (T^8 - 1352T^6 + 1827904T^4 - 2471326208*T^2 + 3341233033216)^2
$53$	\( (T^{8} + \cdots + 10497600000000)^{2} \) (T^8 - 1800T^6 + 3240000T^4 - 5832000000*T^2 + 10497600000000)^2
$59$	\( (T^{8} + \cdots + 1761205026816)^{2} \) (T^8 - 1152T^6 + 1327104T^4 - 1528823808*T^2 + 1761205026816)^2
$61$	\( (T^{8} + \cdots + 80\!\cdots\!16)^{2} \) (T^8 + 9464T^6 + 89567296T^4 + 847664889344*T^2 + 8022300512751616)^2
$67$	\( (T + 42)^{16} \) (T + 42)^16
$71$	\( (T^{8} + \cdots + 320761795710976)^{2} \) (T^8 - 4232T^6 + 17909824T^4 - 75794375168*T^2 + 320761795710976)^2
$73$	\( (T^{8} + \cdots + 983449600000000)^{2} \) (T^8 + 5600T^6 + 31360000T^4 + 175616000000*T^2 + 983449600000000)^2
$79$	\( (T^{8} + 504 T^{6} + \cdots + 64524128256)^{2} \) (T^8 + 504T^6 + 254016T^4 + 128024064*T^2 + 64524128256)^2
$83$	\( (T^{8} - 448 T^{6} + \cdots + 40282095616)^{2} \) (T^8 - 448T^6 + 200704T^4 - 89915392*T^2 + 40282095616)^2
$89$	\( (T^{2} + 3872)^{8} \) (T^2 + 3872)^8
$97$	\( (T^{4} + 74 T^{3} + \cdots + 29986576)^{4} \) (T^4 + 74T^3 + 5476T^2 + 405224*T + 29986576)^4
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\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(-1\)	\(-\beta_{9}\)