LMFDB - Newform orbit 930.2.n.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(481,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("930.481");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.n (of order $5$, degree $4$, 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:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$3$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 16 x^{10} - 6 x^{9} + 161 x^{8} - 180 x^{7} + 1725 x^{6} - 2255 x^{5} + 17635 x^{4} + \cdots + 25 $ x^12 - x^11 + 16x^10 - 6x^9 + 161x^8 - 180x^7 + 1725x^6 - 2255x^5 + 17635x^4 - 18375x^3 + 60975x^2 + 2000x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{7} q^{2} + \beta_{5} q^{3} + ( - \beta_{8} - \beta_{7} - \beta_{5} - 1) q^{4} + q^{5} - q^{6} + (\beta_{8} + \beta_{7} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + \beta_{8} q^{9}+O(q^{10}) $ q - b7 * q^2 + b5 * q^3 + (-b8 - b7 - b5 - 1) * q^4 + q^5 - q^6 + (b8 + b7 + b5 - b3 + 1) * q^7 - b8 * q^8 + b8 * q^9	$ q - \beta_{7} q^{2} + \beta_{5} q^{3} + ( - \beta_{8} - \beta_{7} - \beta_{5} - 1) q^{4} + q^{5} - q^{6} + (\beta_{8} + \beta_{7} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 1) q^{99}+O(q^{100}) $ q - b7 * q^2 + b5 * q^3 + (-b8 - b7 - b5 - 1) * q^4 + q^5 - q^6 + (b8 + b7 + b5 - b3 + 1) * q^7 - b8 * q^8 + b8 * q^9 - b7 * q^10 + (b5 - b3 + 1) * q^11 + b7 * q^12 + (b10 - b8 - b5 - b4 - 1) * q^13 + (b8 - b2) * q^14 + b5 * q^15 + b5 * q^16 + b10 * q^17 - b5 * q^18 + (-b9 + b4 - b1) * q^19 + (-b8 - b7 - b5 - 1) * q^20 + (-b7 + b1) * q^21 + (-b7 - b2 - 1) * q^22 + (-b8 + b6 + b5 - b3) * q^23 + (b8 + b7 + b5 + 1) * q^24 + q^25 + (b10 + b9 + b7 + b5 - b4 + 1) * q^26 + (-b8 - b7 - b5 - 1) * q^27 + b6 * q^28 + (-b11 + b10 - b9) * q^29 - q^30 + (-b9 - b7 + 2b5 - b3 - b2) q^31 - q^32 + (b8 + b5 + b1) * q^33 + (b11 + b9) * q^34 + (b8 + b7 + b5 - b3 + 1) * q^35 + q^36 + (b4 + b3 - b2 - 2) * q^37 + (b11 + b4 - b3) * q^38 + (-b11 + b7 + 1) * q^39 - b8 * q^40 + (-b11 + b10 - 5b7 + b6 + b5 - b4 + b2 + b1) q^41 + (-b8 - b7 - b5 + b3 - 1) * q^42 + (b9 - 2b8 + b6 - b5 - b4 + b2) q^43 + (-b8 + b6 - 1) * q^44 + b8 * q^45 + (b6 + 2b5 - b3 + b1) q^46 + (b10 + b8 + 2b5 - b4 - b3 + b1 + 1) q^47 + b8 * q^48 + (-b10 + b8 - 2b6 + b4 + 1) q^49 - b7 * q^50 + b9 * q^51 + (b9 + b8 + b5 - b4) * q^52 + (-b10 - 2b7 - 2b2 - 2) * q^53 - b8 * q^54 + (b5 - b3 + 1) * q^55 + (b6 + b5 - b3 + b2 + b1 + 1) * q^56 + (b10 + b9 + b6 + b5 - b4 - b3 + b2 + b1) * q^57 + (b11 + b9 + b4) * q^58 + (b10 + 2b8 - 2b6 - b4 + 2) * q^59 + b7 * q^60 + (-b4 - b3 + b2 - 5) * q^61 + (b10 - b8 - b7 + b6 - b2 - 3) * q^62 + (-b6 - b5 + b3 - b2 - b1 - 1) * q^63 + b7 * q^64 + (b10 - b8 - b5 - b4 - 1) * q^65 + (-b5 + b3 - 1) * q^66 + (-3b7 + b6 - 2b5 + b4 + 2b3 - 2b2 + b1) * q^67 - b4 * q^68 + (b8 + b7 + b5 + b2 + b1 + 1) * q^69 + (b8 - b2) * q^70 + (b10 - 6b8) q^71 - b7 * q^72 + (-b11 + b9 - 3b5 - b4 + 2b3 - b2 - b1 - 3) * q^73 + (b11 - b10 + 2b7 + b6 + b5 + b4 + b2) q^74 + b5 * q^75 + (b11 - b2) * q^76 + (-b10 + 2b8 - b6 + 8b5 + b4 - b3 + b1 + 2) * q^77 + (-b10 + b8 + b5 + b4 + 1) * q^78 + (b11 - b10 + 2b8 + b7 - 2b6 - 2b5 + 2b3 + 1) * q^79 + b5 * q^80 + b7 * q^81 + (b9 - 5b8 - 5b7 - 5b5 + b2 + b1 - 5) q^82 + (2b11 - 2b10 - b9 - b8 - b5 + 3b4 - b1) q^83 + (-b8 + b2) * q^84 + b10 * q^85 + (-b11 + 2b5 - b4 - b3 + b2 + b1 + 2) q^86 + (b10 + b9) * q^87 + (b7 + b6 + 2b5 - b3 + b2 + b1 + 1) q^88 + (-2b11 + b8 + b7 + 2b5 - 2b4 - b3 + b2 + b1 + 2) q^89 - b5 * q^90 + (b11 - b10 - 3b8 - b7 - 3b5 + b4 - 3b1) q^91 + (b6 + b5 + b1 - 1) * q^92 + (-b11 + b10 + 2b8 - b4 + b3 + b1 - 1) q^93 + (b10 + b9 - b7 - b5 - b4 + b3 - b2 - 2) * q^94 + (-b9 + b4 - b1) * q^95 - b5 * q^96 + (b11 + 2b9 + 5b8 + 5b7 - 3b5 + b4 + 3b3 - b2 - b1 - 3) q^97 + (-b10 - b9 - b7 - 2b6 - 3b5 + b4 + 2b3 - 2b2 - 2b1 - 2) q^98 + (-b7 - b6 - 2b5 + b3 - b2 - b1 - 1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} - 3 q^{3} - 3 q^{4} + 12 q^{5} - 12 q^{6} - q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) $ 12 * q + 3 * q^2 - 3 * q^3 - 3 * q^4 + 12 * q^5 - 12 * q^6 - q^7 + 3 * q^8 - 3 * q^9	\( 12 q + 3 q^{2} - 3 q^{3} - 3 q^{4} + 12 q^{5} - 12 q^{6} - q^{7} + 3 q^{8} - 3 q^{9} + 3 q^{10} + 5 q^{11} - 3 q^{12} - 5 q^{13} + q^{14} - 3 q^{15} - 3 q^{16} + 3 q^{17} + 3 q^{18} - 2 q^{19} - 3 q^{20} + 4 q^{21} - 5 q^{22} + 3 q^{24} + 12 q^{25} + 10 q^{26} - 3 q^{27} + 4 q^{28} + q^{29} - 12 q^{30} - 6 q^{31} - 12 q^{32} - 5 q^{33} + 2 q^{34} - q^{35} + 12 q^{36} - 14 q^{37} - 3 q^{38} + 10 q^{39} + 3 q^{40} + 15 q^{41} + q^{42} + 10 q^{43} - 5 q^{44} - 3 q^{45} - 5 q^{46} + q^{47} - 3 q^{48} + 3 q^{50} + 3 q^{51} - 5 q^{52} - 13 q^{53} + 3 q^{54} + 5 q^{55} + 6 q^{56} - 2 q^{57} + 4 q^{58} + 11 q^{59} - 3 q^{60} - 70 q^{61} - 19 q^{62} - 6 q^{63} - 3 q^{64} - 5 q^{65} - 5 q^{66} + 38 q^{67} - 2 q^{68} + q^{70} + 21 q^{71} + 3 q^{72} - 14 q^{73} - 11 q^{74} - 3 q^{75} + 3 q^{76} - 14 q^{77} + 5 q^{78} + 5 q^{79} - 3 q^{80} - 3 q^{81} - 15 q^{82} - q^{84} + 3 q^{85} + 10 q^{86} + 6 q^{87} + 3 q^{89} + 3 q^{90} + 16 q^{91} - 10 q^{92} - 11 q^{93} - 6 q^{94} - 2 q^{95} + 3 q^{96} - 35 q^{97} - 10 q^{98}+O(q^{100}) \) 12 * q + 3 * q^2 - 3 * q^3 - 3 * q^4 + 12 * q^5 - 12 * q^6 - q^7 + 3 * q^8 - 3 * q^9 + 3 * q^10 + 5 * q^11 - 3 * q^12 - 5 * q^13 + q^14 - 3 * q^15 - 3 * q^16 + 3 * q^17 + 3 * q^18 - 2 * q^19 - 3 * q^20 + 4 * q^21 - 5 * q^22 + 3 * q^24 + 12 * q^25 + 10 * q^26 - 3 * q^27 + 4 * q^28 + q^29 - 12 * q^30 - 6 * q^31 - 12 * q^32 - 5 * q^33 + 2 * q^34 - q^35 + 12 * q^36 - 14 * q^37 - 3 * q^38 + 10 * q^39 + 3 * q^40 + 15 * q^41 + q^42 + 10 * q^43 - 5 * q^44 - 3 * q^45 - 5 * q^46 + q^47 - 3 * q^48 + 3 * q^50 + 3 * q^51 - 5 * q^52 - 13 * q^53 + 3 * q^54 + 5 * q^55 + 6 * q^56 - 2 * q^57 + 4 * q^58 + 11 * q^59 - 3 * q^60 - 70 * q^61 - 19 * q^62 - 6 * q^63 - 3 * q^64 - 5 * q^65 - 5 * q^66 + 38 * q^67 - 2 * q^68 + q^70 + 21 * q^71 + 3 * q^72 - 14 * q^73 - 11 * q^74 - 3 * q^75 + 3 * q^76 - 14 * q^77 + 5 * q^78 + 5 * q^79 - 3 * q^80 - 3 * q^81 - 15 * q^82 - q^84 + 3 * q^85 + 10 * q^86 + 6 * q^87 + 3 * q^89 + 3 * q^90 + 16 * q^91 - 10 * q^92 - 11 * q^93 - 6 * q^94 - 2 * q^95 + 3 * q^96 - 35 * q^97 - 10 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - x^{11} + 16 x^{10} - 6 x^{9} + 161 x^{8} - 180 x^{7} + 1725 x^{6} - 2255 x^{5} + 17635 x^{4} + \cdots + 25 $ x^12 - x^11 + 16*x^10 - 6*x^9 + 161*x^8 - 180*x^7 + 1725*x^6 - 2255*x^5 + 17635*x^4 - 18375*x^3 + 60975*x^2 + 2000*x + 25 :

$\beta_{1}$	$=$	$ ( 3322281972019 \nu^{11} - 114656979031949 \nu^{10} - 42039345952763 \nu^{9} + \cdots - 13\!\cdots\!50 ) / 40\!\cdots\!30 $ (3322281972019v^11 - 114656979031949v^10 - 42039345952763v^9 - 1446344359381892v^8 - 4528112783420713v^7 - 16341144970954958v^6 - 33087807003119337v^5 - 150045130062088020v^4 - 320325741801216475v^3 - 1212562781876652595v^2 - 4066488446345507205*v - 131773814276113450) / 4027168036955745330
$\beta_{2}$	$=$	$ ( - 112823041844241 \nu^{11} + 329247497650296 \nu^{10} + \cdots - 79\!\cdots\!00 ) / 40\!\cdots\!30 $ (-112823041844241v^11 + 329247497650296v^10 - 1644860785336960v^9 + 3166436229279820v^8 - 12358310003335470v^7 + 45206178197892679v^6 - 200581418861056364v^5 + 475387905458896900v^4 - 2049502436858227190v^3 + 4170956410466929610v^2 - 6378771422167487115*v - 79418675560302400) / 4027168036955745330
$\beta_{3}$	$=$	$ ( - 145525184701524 \nu^{11} + 81175192550164 \nu^{10} + \cdots - 13\!\cdots\!25 ) / 40\!\cdots\!30 $ (-145525184701524v^11 + 81175192550164v^10 - 2131397795696306v^9 + 351234336518491v^8 - 21413987347320571v^7 + 20741106213717582v^6 - 223168834519363377v^5 + 245469958776066705v^4 - 2712776923285910695v^3 + 1818406378451798780v^2 - 6455030843440640165*v - 130511659199335525) / 4027168036955745330
$\beta_{4}$	$=$	$ ( 187465168762881 \nu^{11} + 626582224648512 \nu^{10} + \cdots + 15\!\cdots\!25 ) / 40\!\cdots\!30 $ (187465168762881v^11 + 626582224648512v^10 + 2406078642231698v^9 + 8485330116947397v^8 + 25747569322466333v^7 + 68348535768531023v^6 + 179079646803900835v^5 + 669745210662038795v^4 + 1860639801988947325v^3 + 7190356009182501570v^2 + 233189167335082000*v + 15886129812897084325) / 4027168036955745330
$\beta_{5}$	$=$	$ ( 31\!\cdots\!96 \nu^{11} + \cdots - 25\!\cdots\!15 ) / 40\!\cdots\!30 $ (3176747022412096v^11 - 3289570064256337v^10 + 51157199856243832v^9 - 20705342919809536v^8 + 514622706837627276v^7 - 584172774037512750v^6 + 5525094791858758279v^5 - 7364145954400332844v^4 + 56497321645696209860v^3 - 60422228973680491190v^2 + 197873106102044483210*v - 25277377343295115) / 4027168036955745330
$\beta_{6}$	$=$	$ ( - 31\!\cdots\!96 \nu^{11} + \cdots + 25\!\cdots\!15 ) / 40\!\cdots\!30 $ (-3176747022412096v^11 + 3289570064256337v^10 - 51157199856243832v^9 + 20705342919809536v^8 - 514622706837627276v^7 + 584172774037512750v^6 - 5525094791858758279v^5 + 7364145954400332844v^4 - 56497321645696209860v^3 + 60422228973680491190v^2 - 193845938065088737880*v + 25277377343295115) / 4027168036955745330
$\beta_{7}$	$=$	$ ( - 32\!\cdots\!13 \nu^{11} + \cdots - 64\!\cdots\!75 ) / 40\!\cdots\!30 $ (-3227233225483213v^11 + 3191208800653911v^10 - 51769146933799591v^9 + 18918901688492695v^8 - 520953406836176730v^7 + 576374416026413952v^6 - 5575101240971762564v^5 + 7287911314809457639v^4 - 56992130748017203430v^3 + 58957461773627571845v^2 - 197920533173977390910*v - 6491405462849817175) / 4027168036955745330
$\beta_{8}$	$=$	$ ( - 52\!\cdots\!21 \nu^{11} + \cdots - 39\!\cdots\!35 ) / 40\!\cdots\!30 $ (-5220466367973421v^11 + 5365991552674945v^10 - 83608637080124900v^9 + 33454196003536832v^8 - 840846319580239272v^7 + 961097933582536351v^6 - 9026045590967868807v^5 + 11995320494299427732v^4 - 92308394357987346040v^3 + 98638846434797521570v^2 - 320136343165631144255*v - 3985901892506201835) / 4027168036955745330
$\beta_{9}$	$=$	$ ( - 20\!\cdots\!45 \nu^{11} + \cdots - 25\!\cdots\!50 ) / 40\!\cdots\!30 $ (-20423798056209645v^11 + 21085627500335359v^10 - 328048429442175024v^9 + 130184356242293759v^8 - 3300156092820715519v^7 + 3753906397499170609v^6 - 35428367573615196500v^5 + 46907509583354612425v^4 - 362128821612279045225v^3 + 384094338257193259840v^2 - 1262786767276716280850*v - 25538891027584600850) / 4027168036955745330
$\beta_{10}$	$=$	$ ( 20\!\cdots\!13 \nu^{11} + \cdots + 15\!\cdots\!50 ) / 40\!\cdots\!30 $ (20550414730983713v^11 - 20455879869589488v^10 + 329640044494412703v^9 - 124975585074945293v^8 + 3316491227925739768v^7 - 3716578247124392900v^6 + 35514900285316916445v^5 - 46540443917479469965v^4 + 363332754665948182650v^3 - 379708279637547204655v^2 + 1262928965137764119100*v + 15724276327979534350) / 4027168036955745330
$\beta_{11}$	$=$	$ ( 33\!\cdots\!34 \nu^{11} + \cdots + 25\!\cdots\!00 ) / 40\!\cdots\!30 $ (33316031350330734v^11 - 34370732068070704v^10 + 533491312627074709v^9 - 215260470336265129v^8 + 5364970830225498174v^7 - 6151311119051340505v^6 + 57607217895803319085v^5 - 76679332145286536485v^4 + 589166629757894218850v^3 - 627487295232999333815v^2 + 2043033868985440618875*v + 25437075821649341000) / 4027168036955745330

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

$\nu$	$=$	$ \beta_{6} + \beta_{5} $ b6 + b5
$\nu^{2}$	$=$	$ \beta_{11} + 7\beta_{8} - \beta_{7} - 1 $ b11 + 7*b8 - b7 - 1
$\nu^{3}$	$=$	$ -\beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} - 10\beta_{3} + 1 $ -b11 - b9 - b8 - b7 + b5 - b4 - 10*b3 + 1
$\nu^{4}$	$=$	$ - 3 \beta_{11} + 3 \beta_{10} - 12 \beta_{9} - 7 \beta_{8} + 69 \beta_{7} + \beta_{6} - 6 \beta_{5} + \cdots + 3 \beta_1 $ -3b11 + 3b10 - 12b9 - 7b8 + 69b7 + b6 - 6b5 + 9b4 + b2 + 3b1
$\nu^{5}$	$=$	$ 18 \beta_{10} + 18 \beta_{9} - 32 \beta_{7} - 102 \beta_{6} - 134 \beta_{5} - \beta_{4} + 103 \beta_{3} + \cdots - 6 $ 18b10 + 18b9 - 32b7 - 102b6 - 134b5 - b4 + 103b3 - 103b2 - 102b1 - 6
$\nu^{6}$	$=$	$ 53 \beta_{11} - 137 \beta_{10} + 53 \beta_{9} + 57 \beta_{8} + 63 \beta_{6} + 819 \beta_{5} + 137 \beta_{4} + \cdots + 57 $ 53b11 - 137b10 + 53b9 + 57b8 + 63b6 + 819b5 + 137b4 - 16b3 + 16*b1 + 57
$\nu^{7}$	$=$	$ 253 \beta_{11} - 227 \beta_{10} + 456 \beta_{8} + 392 \beta_{7} + 26 \beta_{6} + 26 \beta_{5} + \cdots + 392 $ 253b11 - 227b10 + 456b8 + 392b7 + 26b6 + 26b5 - 26b3 + 1083b2 + 392
$\nu^{8}$	$=$	$ - 1537 \beta_{11} - 733 \beta_{9} - 7224 \beta_{8} - 7224 \beta_{7} - 7756 \beta_{5} - 1537 \beta_{4} + \cdots - 7756 $ -1537b11 - 733b9 - 7224b8 - 7224b7 - 7756b5 - 1537b4 - 797b3 - 191b2 - 191*b1 - 7756
$\nu^{9}$	$=$	$ - 2812 \beta_{11} + 2812 \beta_{10} - 3258 \beta_{9} + 4347 \beta_{8} + 6796 \beta_{7} + \cdots + 11102 \beta_1 $ -2812b11 + 2812b10 - 3258b9 + 4347b8 + 6796b7 - 461b6 + 3886b5 + 446b4 - 461b2 + 11102b1
$\nu^{10}$	$=$	$ 17172 \beta_{10} + 17172 \beta_{9} + 5462 \beta_{7} - 13758 \beta_{6} - 8296 \beta_{5} - 7829 \beta_{4} + \cdots + 81271 $ 17172b10 + 17172b9 + 5462b7 - 13758b6 - 8296b5 - 7829b4 + 11777b3 - 11777b2 - 13758*b1 + 81271
$\nu^{11}$	$=$	$ 33877 \beta_{11} - 40273 \beta_{10} + 33877 \beta_{9} - 45777 \beta_{8} + 117982 \beta_{6} + \cdots - 45777 $ 33877b11 - 40273b10 + 33877b9 - 45777b8 + 117982b6 + 165016b5 + 40273b4 + 6976b3 - 6976*b1 - 45777

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$.

$n$	$187$	$311$	$871$
$\chi(n)$	$1$	$1$	$\beta_{8}$

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} $

481.1

2.41709	+	1.75612i
−0.0163000	−	0.0118427i
−2.70981	−	1.96879i
1.05776	+	3.25545i
0.716623	+	2.20554i
−0.965367	−	2.97109i
1.05776	−	3.25545i
0.716623	−	2.20554i
−0.965367	+	2.97109i
2.41709	−	1.75612i
−0.0163000	+	0.0118427i
−2.70981	+	1.96879i

0.809017

−

0.587785i

−0.809017

−

0.587785i

0.309017

−

0.951057i

1.00000

−1.00000

−1.23226

3.79252i

−0.309017

−

0.951057i

0.309017

0.951057i

0.809017

−

0.587785i

481.2

0.809017

−

0.587785i

−0.809017

−

0.587785i

0.309017

−

0.951057i

1.00000

−1.00000

−0.302791

0.931895i

−0.309017

−

0.951057i

0.309017

0.951057i

0.809017

−

0.587785i

481.3

0.809017

−

0.587785i

−0.809017

−

0.587785i

0.309017

−

0.951057i

1.00000

−1.00000

0.726038

−

2.23451i

−0.309017

−

0.951057i

0.309017

0.951057i

0.809017

−

0.587785i

721.1

−0.309017

0.951057i

0.309017

0.951057i

−0.809017

−

0.587785i

1.00000

−1.00000

−1.96024

−

1.42420i

0.809017

−

0.587785i

−0.809017

0.587785i

−0.309017

0.951057i

721.2

−0.309017

0.951057i

0.309017

0.951057i

−0.809017

−

0.587785i

1.00000

−1.00000

−1.06713

−

0.775312i

0.809017

−

0.587785i

−0.809017

0.587785i

−0.309017

0.951057i

721.3

−0.309017

0.951057i

0.309017

0.951057i

−0.809017

−

0.587785i

1.00000

−1.00000

3.33638

2.42402i

0.809017

−

0.587785i

−0.809017

0.587785i

−0.309017

0.951057i

841.1

−0.309017

−

0.951057i

0.309017

−

0.951057i

−0.809017

0.587785i

1.00000

−1.00000

−1.96024

1.42420i

0.809017

0.587785i

−0.809017

−

0.587785i

−0.309017

−

0.951057i

841.2

−0.309017

−

0.951057i

0.309017

−

0.951057i

−0.809017

0.587785i

1.00000

−1.00000

−1.06713

0.775312i

0.809017

0.587785i

−0.809017

−

0.587785i

−0.309017

−

0.951057i

841.3

−0.309017

−

0.951057i

0.309017

−

0.951057i

−0.809017

0.587785i

1.00000

−1.00000

3.33638

−

2.42402i

0.809017

0.587785i

−0.809017

−

0.587785i

−0.309017

−

0.951057i

901.1

0.809017

0.587785i

−0.809017

0.587785i

0.309017

0.951057i

1.00000

−1.00000

−1.23226

−

3.79252i

−0.309017

0.951057i

0.309017

−

0.951057i

0.809017

0.587785i

901.2

0.809017

0.587785i

−0.809017

0.587785i

0.309017

0.951057i

1.00000

−1.00000

−0.302791

−

0.931895i

−0.309017

0.951057i

0.309017

−

0.951057i

0.809017

0.587785i

901.3

0.809017

0.587785i

−0.809017

0.587785i

0.309017

0.951057i

1.00000

−1.00000

0.726038

2.23451i

−0.309017

0.951057i

0.309017

−

0.951057i

0.809017

0.587785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.d	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.n.e	✓	12
31.d	even	5	1	inner	930.2.n.e	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.n.e	✓	12	1.a	even	1	1	trivial
930.2.n.e	✓	12	31.d	even	5	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{12} + T_{7}^{11} + 11 T_{7}^{10} - 6 T_{7}^{9} + 132 T_{7}^{8} + 779 T_{7}^{7} + 2921 T_{7}^{6} + \cdots + 14641 $ T7^12 + T7^11 + 11*T7^10 - 6*T7^9 + 132*T7^8 + 779*T7^7 + 2921*T7^6 + 6877*T7^5 + 19194*T7^4 + 34332*T7^3 + 41603*T7^2 + 29645*T7 + 14641 acting on $S_{2}^{\mathrm{new}}(930, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} $ (T^4 - T^3 + T^2 - T + 1)^3
$3$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} $ (T^4 + T^3 + T^2 + T + 1)^3
$5$	$ (T - 1)^{12} $ (T - 1)^12
$7$	$ T^{12} + T^{11} + \cdots + 14641 $ T^12 + T^11 + 11T^10 - 6T^9 + 132T^8 + 779T^7 + 2921T^6 + 6877T^5 + 19194T^4 + 34332T^3 + 41603T^2 + 29645T + 14641
$11$	$ T^{12} - 5 T^{11} + \cdots + 1296 $ T^12 - 5T^11 + 22T^10 - 21T^9 + 119T^8 + 351T^7 + 2224T^6 + 5883T^5 + 10627T^4 + 7302T^3 + 4464T^2 + 2592*T + 1296
$13$	$ T^{12} + 5 T^{11} + \cdots + 1442401 $ T^12 + 5T^11 + 17T^10 - 76T^9 + 514T^8 - 1949T^7 + 25699T^6 - 62997T^5 + 228312T^4 - 575388T^3 + 1242839T^2 - 1468823*T + 1442401
$17$	$ T^{12} - 3 T^{11} + \cdots + 810000 $ T^12 - 3T^11 + 9T^10 - 37T^9 + 606T^8 - 1795T^7 + 14745T^6 - 54375T^5 + 222775T^4 - 555750T^3 + 1008000T^2 - 1080000*T + 810000
$19$	$ T^{12} + 2 T^{11} + \cdots + 164025 $ T^12 + 2T^11 + 19T^10 + 3T^9 + 976T^8 + 5205T^7 + 47070T^6 + 109755T^5 + 822960T^4 + 1877175T^3 + 1986525T^2 + 492075*T + 164025
$23$	$ T^{12} + 43 T^{10} + \cdots + 1296 $ T^12 + 43T^10 - 108T^9 + 659T^8 - 528T^7 + 301T^6 + 4934T^5 + 14197T^4 + 18354T^3 + 15876T^2 + 6696T + 1296
$29$	$ T^{12} - T^{11} + \cdots + 810000 $ T^12 - T^11 + 116T^10 - 141T^9 + 5291T^8 - 4725T^7 + 22580T^6 + 49075T^5 + 14275T^4 - 35250T^3 + 837000T^2 + 1350000T + 810000
$31$	$ T^{12} + \cdots + 887503681 $ T^12 + 6T^11 + 122T^10 + 387T^9 + 5338T^8 + 7042T^7 + 157243T^6 + 218302T^5 + 5129818T^4 + 11529117T^3 + 112669562T^2 + 171774906*T + 887503681
$37$	$ (T^{6} + 7 T^{5} + \cdots + 3659)^{2} $ (T^6 + 7T^5 - 60T^4 - 565T^3 - 800T^2 + 2067*T + 3659)^2
$41$	$ T^{12} - 15 T^{11} + \cdots + 810000 $ T^12 - 15T^11 + 185T^10 - 2205T^9 + 25210T^8 - 84975T^7 + 207675T^6 - 447900T^5 + 1008025T^4 - 1045500T^3 + 1224000T^2 - 1215000*T + 810000
$43$	$ T^{12} + \cdots + 149352841 $ T^12 - 10T^11 + 98T^10 - 502T^9 + 2774T^8 - 13182T^7 + 82281T^6 - 300239T^5 + 1954802T^4 - 10303524T^3 + 41576931T^2 - 86170271*T + 149352841
$47$	$ T^{12} - T^{11} + \cdots + 28772496 $ T^12 - T^11 + 47T^10 - 57T^9 + 1598T^8 - 8937T^7 + 61793T^6 - 398957T^5 + 2614993T^4 - 9396552T^3 + 22452912T^2 - 32473656T + 28772496
$53$	$ T^{12} + 13 T^{11} + \cdots + 95179536 $ T^12 + 13T^11 + 141T^10 + 185T^9 + 890T^8 - 24007T^7 + 709749T^6 - 1310777T^5 + 9303715T^4 + 4538010T^3 + 25426656T^2 + 64857888*T + 95179536
$59$	$ T^{12} + \cdots + 216796176 $ T^12 - 11T^11 + 96T^10 - 59T^9 + 7287T^8 + 50941T^7 + 612056T^6 + 4165203T^5 + 22442059T^4 + 47523738T^3 + 81675108T^2 + 132074280*T + 216796176
$61$	$ (T^{6} + 35 T^{5} + \cdots - 275)^{2} $ (T^6 + 35T^5 + 430T^4 + 2315T^3 + 5430T^2 + 4275*T - 275)^2
$67$	$ (T^{6} - 19 T^{5} + \cdots + 288101)^{2} $ (T^6 - 19T^5 - 165T^4 + 3925T^3 + 840T^2 - 170104*T + 288101)^2
$71$	$ T^{12} - 21 T^{11} + \cdots + 75272976 $ T^12 - 21T^11 + 267T^10 - 2737T^9 + 29898T^8 - 194017T^7 + 897093T^6 - 3405837T^5 + 20824093T^4 - 15667332T^3 + 132517512T^2 - 156740616*T + 75272976
$73$	$ T^{12} + \cdots + 128845201 $ T^12 + 14T^11 + 187T^10 + 463T^9 + 3288T^8 - 46387T^7 + 1067138T^6 - 5242537T^5 + 30160288T^4 - 39501237T^3 + 126144337T^2 + 213954999*T + 128845201
$79$	$ T^{12} - 5 T^{11} + \cdots + 59737441 $ T^12 - 5T^11 + 173T^10 + 1167T^9 + 3364T^8 + 6947T^7 + 1488386T^6 + 17258989T^5 + 129564192T^4 + 492613039T^3 + 919874571T^2 - 119382134*T + 59737441
$83$	$ T^{12} + \cdots + 10608176016 $ T^12 - 3T^10 + 306T^9 + 21394T^8 + 289494T^7 + 2990684T^6 + 19088592T^5 + 125870617T^4 + 590698758T^3 + 1969880184T^2 + 239774688T + 10608176016
$89$	$ T^{12} + \cdots + 26771504400 $ T^12 - 3T^11 + 178T^10 - 1491T^9 + 16865T^8 - 88881T^7 + 820408T^6 - 9320723T^5 + 110257021T^4 - 836256060T^3 + 4543351560T^2 - 14946687000*T + 26771504400
$97$	$ T^{12} + \cdots + 7306041414961 $ T^12 + 35T^11 + 673T^10 + 7082T^9 + 85794T^8 + 2102877T^7 + 59412721T^6 + 1077564529T^5 + 13763718512T^4 + 119243455564T^3 + 738427316631T^2 + 2632959399931*T + 7306041414961
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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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.n (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{2} + \beta_{5} q^{3} + ( - \beta_{8} - \beta_{7} - \beta_{5} - 1) q^{4} + q^{5} - q^{6} + (\beta_{8} + \beta_{7} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + \beta_{8} q^{9}+O(q^{10}) \) q - b7 * q^2 + b5 * q^3 + (-b8 - b7 - b5 - 1) * q^4 + q^5 - q^6 + (b8 + b7 + b5 - b3 + 1) * q^7 - b8 * q^8 + b8 * q^9	\( q - \beta_{7} q^{2} + \beta_{5} q^{3} + ( - \beta_{8} - \beta_{7} - \beta_{5} - 1) q^{4} + q^{5} - q^{6} + (\beta_{8} + \beta_{7} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 1) q^{99}+O(q^{100}) \) q - b7 * q^2 + b5 * q^3 + (-b8 - b7 - b5 - 1) * q^4 + q^5 - q^6 + (b8 + b7 + b5 - b3 + 1) * q^7 - b8 * q^8 + b8 * q^9 - b7 * q^10 + (b5 - b3 + 1) * q^11 + b7 * q^12 + (b10 - b8 - b5 - b4 - 1) * q^13 + (b8 - b2) * q^14 + b5 * q^15 + b5 * q^16 + b10 * q^17 - b5 * q^18 + (-b9 + b4 - b1) * q^19 + (-b8 - b7 - b5 - 1) * q^20 + (-b7 + b1) * q^21 + (-b7 - b2 - 1) * q^22 + (-b8 + b6 + b5 - b3) * q^23 + (b8 + b7 + b5 + 1) * q^24 + q^25 + (b10 + b9 + b7 + b5 - b4 + 1) * q^26 + (-b8 - b7 - b5 - 1) * q^27 + b6 * q^28 + (-b11 + b10 - b9) * q^29 - q^30 + (-b9 - b7 + 2b5 - b3 - b2) q^31 - q^32 + (b8 + b5 + b1) * q^33 + (b11 + b9) * q^34 + (b8 + b7 + b5 - b3 + 1) * q^35 + q^36 + (b4 + b3 - b2 - 2) * q^37 + (b11 + b4 - b3) * q^38 + (-b11 + b7 + 1) * q^39 - b8 * q^40 + (-b11 + b10 - 5b7 + b6 + b5 - b4 + b2 + b1) q^41 + (-b8 - b7 - b5 + b3 - 1) * q^42 + (b9 - 2b8 + b6 - b5 - b4 + b2) q^43 + (-b8 + b6 - 1) * q^44 + b8 * q^45 + (b6 + 2b5 - b3 + b1) q^46 + (b10 + b8 + 2b5 - b4 - b3 + b1 + 1) q^47 + b8 * q^48 + (-b10 + b8 - 2b6 + b4 + 1) q^49 - b7 * q^50 + b9 * q^51 + (b9 + b8 + b5 - b4) * q^52 + (-b10 - 2b7 - 2b2 - 2) * q^53 - b8 * q^54 + (b5 - b3 + 1) * q^55 + (b6 + b5 - b3 + b2 + b1 + 1) * q^56 + (b10 + b9 + b6 + b5 - b4 - b3 + b2 + b1) * q^57 + (b11 + b9 + b4) * q^58 + (b10 + 2b8 - 2b6 - b4 + 2) * q^59 + b7 * q^60 + (-b4 - b3 + b2 - 5) * q^61 + (b10 - b8 - b7 + b6 - b2 - 3) * q^62 + (-b6 - b5 + b3 - b2 - b1 - 1) * q^63 + b7 * q^64 + (b10 - b8 - b5 - b4 - 1) * q^65 + (-b5 + b3 - 1) * q^66 + (-3b7 + b6 - 2b5 + b4 + 2b3 - 2b2 + b1) * q^67 - b4 * q^68 + (b8 + b7 + b5 + b2 + b1 + 1) * q^69 + (b8 - b2) * q^70 + (b10 - 6b8) q^71 - b7 * q^72 + (-b11 + b9 - 3b5 - b4 + 2b3 - b2 - b1 - 3) * q^73 + (b11 - b10 + 2b7 + b6 + b5 + b4 + b2) q^74 + b5 * q^75 + (b11 - b2) * q^76 + (-b10 + 2b8 - b6 + 8b5 + b4 - b3 + b1 + 2) * q^77 + (-b10 + b8 + b5 + b4 + 1) * q^78 + (b11 - b10 + 2b8 + b7 - 2b6 - 2b5 + 2b3 + 1) * q^79 + b5 * q^80 + b7 * q^81 + (b9 - 5b8 - 5b7 - 5b5 + b2 + b1 - 5) q^82 + (2b11 - 2b10 - b9 - b8 - b5 + 3b4 - b1) q^83 + (-b8 + b2) * q^84 + b10 * q^85 + (-b11 + 2b5 - b4 - b3 + b2 + b1 + 2) q^86 + (b10 + b9) * q^87 + (b7 + b6 + 2b5 - b3 + b2 + b1 + 1) q^88 + (-2b11 + b8 + b7 + 2b5 - 2b4 - b3 + b2 + b1 + 2) q^89 - b5 * q^90 + (b11 - b10 - 3b8 - b7 - 3b5 + b4 - 3b1) q^91 + (b6 + b5 + b1 - 1) * q^92 + (-b11 + b10 + 2b8 - b4 + b3 + b1 - 1) q^93 + (b10 + b9 - b7 - b5 - b4 + b3 - b2 - 2) * q^94 + (-b9 + b4 - b1) * q^95 - b5 * q^96 + (b11 + 2b9 + 5b8 + 5b7 - 3b5 + b4 + 3b3 - b2 - b1 - 3) q^97 + (-b10 - b9 - b7 - 2b6 - 3b5 + b4 + 2b3 - 2b2 - 2b1 - 2) q^98 + (-b7 - b6 - 2b5 + b3 - b2 - b1 - 1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} - 3 q^{3} - 3 q^{4} + 12 q^{5} - 12 q^{6} - q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) 12 * q + 3 * q^2 - 3 * q^3 - 3 * q^4 + 12 * q^5 - 12 * q^6 - q^7 + 3 * q^8 - 3 * q^9	\( 12 q + 3 q^{2} - 3 q^{3} - 3 q^{4} + 12 q^{5} - 12 q^{6} - q^{7} + 3 q^{8} - 3 q^{9} + 3 q^{10} + 5 q^{11} - 3 q^{12} - 5 q^{13} + q^{14} - 3 q^{15} - 3 q^{16} + 3 q^{17} + 3 q^{18} - 2 q^{19} - 3 q^{20} + 4 q^{21} - 5 q^{22} + 3 q^{24} + 12 q^{25} + 10 q^{26} - 3 q^{27} + 4 q^{28} + q^{29} - 12 q^{30} - 6 q^{31} - 12 q^{32} - 5 q^{33} + 2 q^{34} - q^{35} + 12 q^{36} - 14 q^{37} - 3 q^{38} + 10 q^{39} + 3 q^{40} + 15 q^{41} + q^{42} + 10 q^{43} - 5 q^{44} - 3 q^{45} - 5 q^{46} + q^{47} - 3 q^{48} + 3 q^{50} + 3 q^{51} - 5 q^{52} - 13 q^{53} + 3 q^{54} + 5 q^{55} + 6 q^{56} - 2 q^{57} + 4 q^{58} + 11 q^{59} - 3 q^{60} - 70 q^{61} - 19 q^{62} - 6 q^{63} - 3 q^{64} - 5 q^{65} - 5 q^{66} + 38 q^{67} - 2 q^{68} + q^{70} + 21 q^{71} + 3 q^{72} - 14 q^{73} - 11 q^{74} - 3 q^{75} + 3 q^{76} - 14 q^{77} + 5 q^{78} + 5 q^{79} - 3 q^{80} - 3 q^{81} - 15 q^{82} - q^{84} + 3 q^{85} + 10 q^{86} + 6 q^{87} + 3 q^{89} + 3 q^{90} + 16 q^{91} - 10 q^{92} - 11 q^{93} - 6 q^{94} - 2 q^{95} + 3 q^{96} - 35 q^{97} - 10 q^{98}+O(q^{100}) \) 12 * q + 3 * q^2 - 3 * q^3 - 3 * q^4 + 12 * q^5 - 12 * q^6 - q^7 + 3 * q^8 - 3 * q^9 + 3 * q^10 + 5 * q^11 - 3 * q^12 - 5 * q^13 + q^14 - 3 * q^15 - 3 * q^16 + 3 * q^17 + 3 * q^18 - 2 * q^19 - 3 * q^20 + 4 * q^21 - 5 * q^22 + 3 * q^24 + 12 * q^25 + 10 * q^26 - 3 * q^27 + 4 * q^28 + q^29 - 12 * q^30 - 6 * q^31 - 12 * q^32 - 5 * q^33 + 2 * q^34 - q^35 + 12 * q^36 - 14 * q^37 - 3 * q^38 + 10 * q^39 + 3 * q^40 + 15 * q^41 + q^42 + 10 * q^43 - 5 * q^44 - 3 * q^45 - 5 * q^46 + q^47 - 3 * q^48 + 3 * q^50 + 3 * q^51 - 5 * q^52 - 13 * q^53 + 3 * q^54 + 5 * q^55 + 6 * q^56 - 2 * q^57 + 4 * q^58 + 11 * q^59 - 3 * q^60 - 70 * q^61 - 19 * q^62 - 6 * q^63 - 3 * q^64 - 5 * q^65 - 5 * q^66 + 38 * q^67 - 2 * q^68 + q^70 + 21 * q^71 + 3 * q^72 - 14 * q^73 - 11 * q^74 - 3 * q^75 + 3 * q^76 - 14 * q^77 + 5 * q^78 + 5 * q^79 - 3 * q^80 - 3 * q^81 - 15 * q^82 - q^84 + 3 * q^85 + 10 * q^86 + 6 * q^87 + 3 * q^89 + 3 * q^90 + 16 * q^91 - 10 * q^92 - 11 * q^93 - 6 * q^94 - 2 * q^95 + 3 * q^96 - 35 * q^97 - 10 * q^98

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	930.2.n.e

Level	$930$
Weight	$2$
Character orbit	930.n
Analytic conductor	$7.426$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(7.42608738798\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + 16 x^{10} - 6 x^{9} + 161 x^{8} - 180 x^{7} + 1725 x^{6} - 2255 x^{5} + 17635 x^{4} + \cdots + 25 \) x^12 - x^11 + 16x^10 - 6x^9 + 161x^8 - 180x^7 + 1725x^6 - 2255x^5 + 17635x^4 - 18375x^3 + 60975x^2 + 2000x + 25
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 5 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

\(\beta_{1}\)	\(=\)	\( ( 3322281972019 \nu^{11} - 114656979031949 \nu^{10} - 42039345952763 \nu^{9} + \cdots - 13\!\cdots\!50 ) / 40\!\cdots\!30 \) (3322281972019v^11 - 114656979031949v^10 - 42039345952763v^9 - 1446344359381892v^8 - 4528112783420713v^7 - 16341144970954958v^6 - 33087807003119337v^5 - 150045130062088020v^4 - 320325741801216475v^3 - 1212562781876652595v^2 - 4066488446345507205*v - 131773814276113450) / 4027168036955745330
\(\beta_{2}\)	\(=\)	\( ( - 112823041844241 \nu^{11} + 329247497650296 \nu^{10} + \cdots - 79\!\cdots\!00 ) / 40\!\cdots\!30 \) (-112823041844241v^11 + 329247497650296v^10 - 1644860785336960v^9 + 3166436229279820v^8 - 12358310003335470v^7 + 45206178197892679v^6 - 200581418861056364v^5 + 475387905458896900v^4 - 2049502436858227190v^3 + 4170956410466929610v^2 - 6378771422167487115*v - 79418675560302400) / 4027168036955745330
\(\beta_{3}\)	\(=\)	\( ( - 145525184701524 \nu^{11} + 81175192550164 \nu^{10} + \cdots - 13\!\cdots\!25 ) / 40\!\cdots\!30 \) (-145525184701524v^11 + 81175192550164v^10 - 2131397795696306v^9 + 351234336518491v^8 - 21413987347320571v^7 + 20741106213717582v^6 - 223168834519363377v^5 + 245469958776066705v^4 - 2712776923285910695v^3 + 1818406378451798780v^2 - 6455030843440640165*v - 130511659199335525) / 4027168036955745330
\(\beta_{4}\)	\(=\)	\( ( 187465168762881 \nu^{11} + 626582224648512 \nu^{10} + \cdots + 15\!\cdots\!25 ) / 40\!\cdots\!30 \) (187465168762881v^11 + 626582224648512v^10 + 2406078642231698v^9 + 8485330116947397v^8 + 25747569322466333v^7 + 68348535768531023v^6 + 179079646803900835v^5 + 669745210662038795v^4 + 1860639801988947325v^3 + 7190356009182501570v^2 + 233189167335082000*v + 15886129812897084325) / 4027168036955745330
\(\beta_{5}\)	\(=\)	\( ( 31\!\cdots\!96 \nu^{11} + \cdots - 25\!\cdots\!15 ) / 40\!\cdots\!30 \) (3176747022412096v^11 - 3289570064256337v^10 + 51157199856243832v^9 - 20705342919809536v^8 + 514622706837627276v^7 - 584172774037512750v^6 + 5525094791858758279v^5 - 7364145954400332844v^4 + 56497321645696209860v^3 - 60422228973680491190v^2 + 197873106102044483210*v - 25277377343295115) / 4027168036955745330
\(\beta_{6}\)	\(=\)	\( ( - 31\!\cdots\!96 \nu^{11} + \cdots + 25\!\cdots\!15 ) / 40\!\cdots\!30 \) (-3176747022412096v^11 + 3289570064256337v^10 - 51157199856243832v^9 + 20705342919809536v^8 - 514622706837627276v^7 + 584172774037512750v^6 - 5525094791858758279v^5 + 7364145954400332844v^4 - 56497321645696209860v^3 + 60422228973680491190v^2 - 193845938065088737880*v + 25277377343295115) / 4027168036955745330
\(\beta_{7}\)	\(=\)	\( ( - 32\!\cdots\!13 \nu^{11} + \cdots - 64\!\cdots\!75 ) / 40\!\cdots\!30 \) (-3227233225483213v^11 + 3191208800653911v^10 - 51769146933799591v^9 + 18918901688492695v^8 - 520953406836176730v^7 + 576374416026413952v^6 - 5575101240971762564v^5 + 7287911314809457639v^4 - 56992130748017203430v^3 + 58957461773627571845v^2 - 197920533173977390910*v - 6491405462849817175) / 4027168036955745330
\(\beta_{8}\)	\(=\)	\( ( - 52\!\cdots\!21 \nu^{11} + \cdots - 39\!\cdots\!35 ) / 40\!\cdots\!30 \) (-5220466367973421v^11 + 5365991552674945v^10 - 83608637080124900v^9 + 33454196003536832v^8 - 840846319580239272v^7 + 961097933582536351v^6 - 9026045590967868807v^5 + 11995320494299427732v^4 - 92308394357987346040v^3 + 98638846434797521570v^2 - 320136343165631144255*v - 3985901892506201835) / 4027168036955745330
\(\beta_{9}\)	\(=\)	\( ( - 20\!\cdots\!45 \nu^{11} + \cdots - 25\!\cdots\!50 ) / 40\!\cdots\!30 \) (-20423798056209645v^11 + 21085627500335359v^10 - 328048429442175024v^9 + 130184356242293759v^8 - 3300156092820715519v^7 + 3753906397499170609v^6 - 35428367573615196500v^5 + 46907509583354612425v^4 - 362128821612279045225v^3 + 384094338257193259840v^2 - 1262786767276716280850*v - 25538891027584600850) / 4027168036955745330
\(\beta_{10}\)	\(=\)	\( ( 20\!\cdots\!13 \nu^{11} + \cdots + 15\!\cdots\!50 ) / 40\!\cdots\!30 \) (20550414730983713v^11 - 20455879869589488v^10 + 329640044494412703v^9 - 124975585074945293v^8 + 3316491227925739768v^7 - 3716578247124392900v^6 + 35514900285316916445v^5 - 46540443917479469965v^4 + 363332754665948182650v^3 - 379708279637547204655v^2 + 1262928965137764119100*v + 15724276327979534350) / 4027168036955745330
\(\beta_{11}\)	\(=\)	\( ( 33\!\cdots\!34 \nu^{11} + \cdots + 25\!\cdots\!00 ) / 40\!\cdots\!30 \) (33316031350330734v^11 - 34370732068070704v^10 + 533491312627074709v^9 - 215260470336265129v^8 + 5364970830225498174v^7 - 6151311119051340505v^6 + 57607217895803319085v^5 - 76679332145286536485v^4 + 589166629757894218850v^3 - 627487295232999333815v^2 + 2043033868985440618875*v + 25437075821649341000) / 4027168036955745330

\(\nu\)	\(=\)	\( \beta_{6} + \beta_{5} \) b6 + b5
\(\nu^{2}\)	\(=\)	\( \beta_{11} + 7\beta_{8} - \beta_{7} - 1 \) b11 + 7*b8 - b7 - 1
\(\nu^{3}\)	\(=\)	\( -\beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} - 10\beta_{3} + 1 \) -b11 - b9 - b8 - b7 + b5 - b4 - 10*b3 + 1
\(\nu^{4}\)	\(=\)	\( - 3 \beta_{11} + 3 \beta_{10} - 12 \beta_{9} - 7 \beta_{8} + 69 \beta_{7} + \beta_{6} - 6 \beta_{5} + \cdots + 3 \beta_1 \) -3b11 + 3b10 - 12b9 - 7b8 + 69b7 + b6 - 6b5 + 9b4 + b2 + 3b1
\(\nu^{5}\)	\(=\)	\( 18 \beta_{10} + 18 \beta_{9} - 32 \beta_{7} - 102 \beta_{6} - 134 \beta_{5} - \beta_{4} + 103 \beta_{3} + \cdots - 6 \) 18b10 + 18b9 - 32b7 - 102b6 - 134b5 - b4 + 103b3 - 103b2 - 102b1 - 6
\(\nu^{6}\)	\(=\)	\( 53 \beta_{11} - 137 \beta_{10} + 53 \beta_{9} + 57 \beta_{8} + 63 \beta_{6} + 819 \beta_{5} + 137 \beta_{4} + \cdots + 57 \) 53b11 - 137b10 + 53b9 + 57b8 + 63b6 + 819b5 + 137b4 - 16b3 + 16*b1 + 57
\(\nu^{7}\)	\(=\)	\( 253 \beta_{11} - 227 \beta_{10} + 456 \beta_{8} + 392 \beta_{7} + 26 \beta_{6} + 26 \beta_{5} + \cdots + 392 \) 253b11 - 227b10 + 456b8 + 392b7 + 26b6 + 26b5 - 26b3 + 1083b2 + 392
\(\nu^{8}\)	\(=\)	\( - 1537 \beta_{11} - 733 \beta_{9} - 7224 \beta_{8} - 7224 \beta_{7} - 7756 \beta_{5} - 1537 \beta_{4} + \cdots - 7756 \) -1537b11 - 733b9 - 7224b8 - 7224b7 - 7756b5 - 1537b4 - 797b3 - 191b2 - 191*b1 - 7756
\(\nu^{9}\)	\(=\)	\( - 2812 \beta_{11} + 2812 \beta_{10} - 3258 \beta_{9} + 4347 \beta_{8} + 6796 \beta_{7} + \cdots + 11102 \beta_1 \) -2812b11 + 2812b10 - 3258b9 + 4347b8 + 6796b7 - 461b6 + 3886b5 + 446b4 - 461b2 + 11102b1
\(\nu^{10}\)	\(=\)	\( 17172 \beta_{10} + 17172 \beta_{9} + 5462 \beta_{7} - 13758 \beta_{6} - 8296 \beta_{5} - 7829 \beta_{4} + \cdots + 81271 \) 17172b10 + 17172b9 + 5462b7 - 13758b6 - 8296b5 - 7829b4 + 11777b3 - 11777b2 - 13758*b1 + 81271
\(\nu^{11}\)	\(=\)	\( 33877 \beta_{11} - 40273 \beta_{10} + 33877 \beta_{9} - 45777 \beta_{8} + 117982 \beta_{6} + \cdots - 45777 \) 33877b11 - 40273b10 + 33877b9 - 45777b8 + 117982b6 + 165016b5 + 40273b4 + 6976b3 - 6976*b1 - 45777

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(1\)	\(1\)	\(\beta_{8}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 481.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) (T^4 - T^3 + T^2 - T + 1)^3
$3$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) (T^4 + T^3 + T^2 + T + 1)^3
$5$	\( (T - 1)^{12} \) (T - 1)^12
$7$	\( T^{12} + T^{11} + \cdots + 14641 \) T^12 + T^11 + 11T^10 - 6T^9 + 132T^8 + 779T^7 + 2921T^6 + 6877T^5 + 19194T^4 + 34332T^3 + 41603T^2 + 29645T + 14641
$11$	\( T^{12} - 5 T^{11} + \cdots + 1296 \) T^12 - 5T^11 + 22T^10 - 21T^9 + 119T^8 + 351T^7 + 2224T^6 + 5883T^5 + 10627T^4 + 7302T^3 + 4464T^2 + 2592*T + 1296
$13$	\( T^{12} + 5 T^{11} + \cdots + 1442401 \) T^12 + 5T^11 + 17T^10 - 76T^9 + 514T^8 - 1949T^7 + 25699T^6 - 62997T^5 + 228312T^4 - 575388T^3 + 1242839T^2 - 1468823*T + 1442401
$17$	\( T^{12} - 3 T^{11} + \cdots + 810000 \) T^12 - 3T^11 + 9T^10 - 37T^9 + 606T^8 - 1795T^7 + 14745T^6 - 54375T^5 + 222775T^4 - 555750T^3 + 1008000T^2 - 1080000*T + 810000
$19$	\( T^{12} + 2 T^{11} + \cdots + 164025 \) T^12 + 2T^11 + 19T^10 + 3T^9 + 976T^8 + 5205T^7 + 47070T^6 + 109755T^5 + 822960T^4 + 1877175T^3 + 1986525T^2 + 492075*T + 164025
$23$	\( T^{12} + 43 T^{10} + \cdots + 1296 \) T^12 + 43T^10 - 108T^9 + 659T^8 - 528T^7 + 301T^6 + 4934T^5 + 14197T^4 + 18354T^3 + 15876T^2 + 6696T + 1296
$29$	\( T^{12} - T^{11} + \cdots + 810000 \) T^12 - T^11 + 116T^10 - 141T^9 + 5291T^8 - 4725T^7 + 22580T^6 + 49075T^5 + 14275T^4 - 35250T^3 + 837000T^2 + 1350000T + 810000
$31$	\( T^{12} + \cdots + 887503681 \) T^12 + 6T^11 + 122T^10 + 387T^9 + 5338T^8 + 7042T^7 + 157243T^6 + 218302T^5 + 5129818T^4 + 11529117T^3 + 112669562T^2 + 171774906*T + 887503681
$37$	\( (T^{6} + 7 T^{5} + \cdots + 3659)^{2} \) (T^6 + 7T^5 - 60T^4 - 565T^3 - 800T^2 + 2067*T + 3659)^2
$41$	\( T^{12} - 15 T^{11} + \cdots + 810000 \) T^12 - 15T^11 + 185T^10 - 2205T^9 + 25210T^8 - 84975T^7 + 207675T^6 - 447900T^5 + 1008025T^4 - 1045500T^3 + 1224000T^2 - 1215000*T + 810000
$43$	\( T^{12} + \cdots + 149352841 \) T^12 - 10T^11 + 98T^10 - 502T^9 + 2774T^8 - 13182T^7 + 82281T^6 - 300239T^5 + 1954802T^4 - 10303524T^3 + 41576931T^2 - 86170271*T + 149352841
$47$	\( T^{12} - T^{11} + \cdots + 28772496 \) T^12 - T^11 + 47T^10 - 57T^9 + 1598T^8 - 8937T^7 + 61793T^6 - 398957T^5 + 2614993T^4 - 9396552T^3 + 22452912T^2 - 32473656T + 28772496
$53$	\( T^{12} + 13 T^{11} + \cdots + 95179536 \) T^12 + 13T^11 + 141T^10 + 185T^9 + 890T^8 - 24007T^7 + 709749T^6 - 1310777T^5 + 9303715T^4 + 4538010T^3 + 25426656T^2 + 64857888*T + 95179536
$59$	\( T^{12} + \cdots + 216796176 \) T^12 - 11T^11 + 96T^10 - 59T^9 + 7287T^8 + 50941T^7 + 612056T^6 + 4165203T^5 + 22442059T^4 + 47523738T^3 + 81675108T^2 + 132074280*T + 216796176
$61$	\( (T^{6} + 35 T^{5} + \cdots - 275)^{2} \) (T^6 + 35T^5 + 430T^4 + 2315T^3 + 5430T^2 + 4275*T - 275)^2
$67$	\( (T^{6} - 19 T^{5} + \cdots + 288101)^{2} \) (T^6 - 19T^5 - 165T^4 + 3925T^3 + 840T^2 - 170104*T + 288101)^2
$71$	\( T^{12} - 21 T^{11} + \cdots + 75272976 \) T^12 - 21T^11 + 267T^10 - 2737T^9 + 29898T^8 - 194017T^7 + 897093T^6 - 3405837T^5 + 20824093T^4 - 15667332T^3 + 132517512T^2 - 156740616*T + 75272976
$73$	\( T^{12} + \cdots + 128845201 \) T^12 + 14T^11 + 187T^10 + 463T^9 + 3288T^8 - 46387T^7 + 1067138T^6 - 5242537T^5 + 30160288T^4 - 39501237T^3 + 126144337T^2 + 213954999*T + 128845201
$79$	\( T^{12} - 5 T^{11} + \cdots + 59737441 \) T^12 - 5T^11 + 173T^10 + 1167T^9 + 3364T^8 + 6947T^7 + 1488386T^6 + 17258989T^5 + 129564192T^4 + 492613039T^3 + 919874571T^2 - 119382134*T + 59737441
$83$	\( T^{12} + \cdots + 10608176016 \) T^12 - 3T^10 + 306T^9 + 21394T^8 + 289494T^7 + 2990684T^6 + 19088592T^5 + 125870617T^4 + 590698758T^3 + 1969880184T^2 + 239774688T + 10608176016
$89$	\( T^{12} + \cdots + 26771504400 \) T^12 - 3T^11 + 178T^10 - 1491T^9 + 16865T^8 - 88881T^7 + 820408T^6 - 9320723T^5 + 110257021T^4 - 836256060T^3 + 4543351560T^2 - 14946687000*T + 26771504400
$97$	\( T^{12} + \cdots + 7306041414961 \) T^12 + 35T^11 + 673T^10 + 7082T^9 + 85794T^8 + 2102877T^7 + 59412721T^6 + 1077564529T^5 + 13763718512T^4 + 119243455564T^3 + 738427316631T^2 + 2632959399931*T + 7306041414961
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