LMFDB - Newform orbit 510.2.m.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [510,2,Mod(259,510)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(510, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("510.259");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 510 = 2 \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	510.m (of order $4$, degree $2$, 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:	$4.07237050309$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
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} + 2x^{10} - 20x^{9} + 2x^{8} + 76x^{7} + 74x^{6} + 116x^{5} + 304x^{4} + 276x^{3} + 104x^{2} + 16x + 2 $ x^12 + 2x^10 - 20x^9 + 2x^8 + 76x^7 + 74x^6 + 116x^5 + 304x^4 + 276x^3 + 104x^2 + 16x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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 - q^{2} - \beta_{7} q^{3} + q^{4} - \beta_{2} q^{5} + \beta_{7} q^{6} + (\beta_{6} + \beta_{4}) q^{7} - q^{8} - \beta_1 q^{9}+O(q^{10}) $ q - q^2 - b7 * q^3 + q^4 - b2 * q^5 + b7 * q^6 + (b6 + b4) * q^7 - q^8 - b1 * q^9	$ q - q^{2} - \beta_{7} q^{3} + q^{4} - \beta_{2} q^{5} + \beta_{7} q^{6} + (\beta_{6} + \beta_{4}) q^{7} - q^{8} - \beta_1 q^{9} + \beta_{2} q^{10} + ( - 2 \beta_1 + 2) q^{11} - \beta_{7} q^{12} + ( - \beta_{11} + \beta_{9} + \cdots + 2 \beta_1) q^{13}+ \cdots + ( - 2 \beta_1 - 2) q^{99}+O(q^{100}) $ q - q^2 - b7 * q^3 + q^4 - b2 * q^5 + b7 * q^6 + (b6 + b4) * q^7 - q^8 - b1 * q^9 + b2 * q^10 + (-2b1 + 2) q^11 - b7 * q^12 + (-b11 + b9 - b7 + b6 - b5 - b3 + 2b1) q^13 + (-b6 - b4) * q^14 - b6 * q^15 + q^16 + (b10 + b8 - b7 + b4 - b3 + b2 - b1) * q^17 + b1 * q^18 + (b9 + b8 - 2b7 + b6 - b4 - 2b3 + 2b1) q^19 - b2 * q^20 + (b11 - b5) * q^21 + (2b1 - 2) q^22 + (b11 + b10 - 3b1 - 3) q^23 + b7 * q^24 + (-b11 + b9 - b8 - 3b7 - 2b4 + b3 - b2 + 2b1 + 1) q^25 + (b11 - b9 + b7 - b6 + b5 + b3 - 2b1) q^26 - b3 * q^27 + (b6 + b4) * q^28 + (b11 + b10 - b9 - b8 + b6 + b4 - 2b3 - 2b1 - 2) * q^29 + b6 * q^30 + (-2b11 - 2b10 + b9 + b8 - b6 + b5 - b4 + 2b3 - b2 + b1 + 1) q^31 - q^32 + (-2b7 - 2b3) * q^33 + (-b10 - b8 + b7 - b4 + b3 - b2 + b1) * q^34 + (-b10 + b8 - 4b7 + b6 - 2b5 - 2b3 - b2 + 3b1 - 1) * q^35 - b1 * q^36 + (-b9 + b8 + 2b7 + b6 - b5 - b4 - b2) q^37 + (-b9 - b8 + 2b7 - b6 + b4 + 2b3 - 2b1) q^38 + (b11 + b10 - b9 - b8 + 2b3 - b1 - 1) q^39 + b2 * q^40 + (-b11 + b10 + b9 - b8 - 4b7 - 2b5 - 2b2 + 2b1 - 2) * q^41 + (-b11 + b5) * q^42 + (b11 + b8 + 3b7 - b5 + b4 - 3b3 + 2) * q^43 + (-2b1 + 2) q^44 - b11 * q^45 + (-b11 - b10 + 3b1 + 3) q^46 + (b11 - b10 - 2b7 + b5 - 2b3 - b2 + 2b1) q^47 - b7 * q^48 + (-b11 + b10 + 3b9 - b8 - 4b7 + 3b6 - b5 + b4 - 4b3 + b2 + b1) * q^49 + (b11 - b9 + b8 + 3b7 + 2b4 - b3 + b2 - 2b1 - 1) q^50 + (b6 - b5 + b4 - b3 - b2 - b1 - 1) * q^51 + (-b11 + b9 - b7 + b6 - b5 - b3 + 2b1) q^52 + (-b11 + b10 - 2b8 - 2b7 + b5 - 2b4 + 2b3 - b2 + 2) * q^53 + b3 * q^54 + (-2b11 - 2b2) * q^55 + (-b6 - b4) * q^56 + (b11 + b10 + b5 + 2b3 - b2 - 2b1 - 2) * q^57 + (-b11 - b10 + b9 + b8 - b6 - b4 + 2b3 + 2b1 + 2) * q^58 + (b10 + b8 + b7 - b4 + b3 + b2) * q^59 - b6 * q^60 + (-b11 + b10 + b9 - b8 - 6b7 + b6 - 2b5 - b4 - 2b2 + 2b1 - 2) * q^61 + (2b11 + 2b10 - b9 - b8 + b6 - b5 + b4 - 2b3 + b2 - b1 - 1) q^62 + (-b9 + b8) * q^63 + q^64 + (b11 - b10 + 2b9 + b6 - 2b5 + b4 - 3b1 + 1) q^65 + (2b7 + 2b3) * q^66 + (2b11 + 3b10 - b8 + b7 + 2b5 + b4 + b3 + 3b2 - 4b1) q^67 + (b10 + b8 - b7 + b4 - b3 + b2 - b1) * q^68 + (b8 + 3b7 + b4 - 3b3) * q^69 + (b10 - b8 + 4b7 - b6 + 2b5 + 2b3 + b2 - 3b1 + 1) * q^70 + (-b11 - b10 - b9 - b8 - 2b6 - b5 - 2b4 - 2b3 + b2 + 2b1 + 2) * q^71 + b1 * q^72 + (2b11 - 2b10 - 2b7 + 3b6 - 3b5 - 3b4 - 3b2 + 2b1 - 2) * q^73 + (b9 - b8 - 2b7 - b6 + b5 + b4 + b2) q^74 + (b10 - b8 - b7 - b6 + 2b5 + 2b3 + b2 - 3b1 + 1) q^75 + (b9 + b8 - 2b7 + b6 - b4 - 2b3 + 2b1) q^76 + (-2b9 + 2b8 + 2b6 + 2b4) * q^77 + (-b11 - b10 + b9 + b8 - 2b3 + b1 + 1) q^78 + (b11 - b10 - 3b9 + 3b8 + 2b7 - b6 + 2b5 + b4 + 2b2 + b1 - 1) q^79 - b2 * q^80 - q^81 + (b11 - b10 - b9 + b8 + 4b7 + 2b5 + 2b2 - 2b1 + 2) * q^82 + (-2b11 - 2b10 + 3b9 - b8 - 2b7 - 3b6 + 2b5 - b4 + 2b3 + 2b2 + 6) * q^83 + (b11 - b5) * q^84 + (-b11 - 2b10 - b9 + b8 - 2b6 + b5 + 2b4 + b2 - b1 + 7) q^85 + (-b11 - b8 - 3b7 + b5 - b4 + 3b3 - 2) * q^86 + (b11 - b10 + b8 + 2b7 - b5 + b4 - 2b3 + b2 - 2) * q^87 + (2b1 - 2) q^88 + (b11 + b10 - b9 + b8 - 2b7 + b6 - b5 + b4 + 2b3 - b2 - 8) * q^89 + b11 * q^90 + (b11 - b10 + 3b9 - 3b8 + 2b7 - b6 + b5 + b4 + b2) q^91 + (b11 + b10 - 3b1 - 3) q^92 + (-b11 + b10 + b9 - 2b8 - b7 - b6 + b5 - 2b4 + b3 - b2 + 2) * q^93 + (-b11 + b10 + 2b7 - b5 + 2b3 + b2 - 2b1) q^94 + (b11 - 3b10 + 8b7 - 2b6 - b5 + 4b3 - b2 + 4b1 + 2) q^95 + b7 * q^96 + (-2b9 + 2b8 + 2b7 - b6 + 3b5 + b4 + 3b2) q^97 + (b11 - b10 - 3b9 + b8 + 4b7 - 3b6 + b5 - b4 + 4b3 - b2 - b1) * q^98 + (-2b1 - 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} + 4 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 + 4 * q^5 + 4 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} + 4 q^{5} + 4 q^{7} - 12 q^{8} - 4 q^{10} + 24 q^{11} - 4 q^{14} - 4 q^{15} + 12 q^{16} + 4 q^{20} + 8 q^{21} - 24 q^{22} - 28 q^{23} + 8 q^{25} + 4 q^{28} - 8 q^{29} + 4 q^{30} - 12 q^{31} - 12 q^{32} + 16 q^{37} - 4 q^{40} - 12 q^{41} - 8 q^{42} + 32 q^{43} + 24 q^{44} - 4 q^{45} + 28 q^{46} - 8 q^{50} + 24 q^{53} - 4 q^{56} - 16 q^{57} + 8 q^{58} - 4 q^{60} - 8 q^{61} + 12 q^{62} + 4 q^{63} + 12 q^{64} + 16 q^{65} + 12 q^{71} + 12 q^{73} - 16 q^{74} + 16 q^{77} - 20 q^{79} + 4 q^{80} - 12 q^{81} + 12 q^{82} + 16 q^{83} + 8 q^{84} + 60 q^{85} - 32 q^{86} - 24 q^{87} - 24 q^{88} - 72 q^{89} + 4 q^{90} - 24 q^{91} - 28 q^{92} + 16 q^{93} + 16 q^{95} - 20 q^{97} - 24 q^{99}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 + 4 * q^5 + 4 * q^7 - 12 * q^8 - 4 * q^10 + 24 * q^11 - 4 * q^14 - 4 * q^15 + 12 * q^16 + 4 * q^20 + 8 * q^21 - 24 * q^22 - 28 * q^23 + 8 * q^25 + 4 * q^28 - 8 * q^29 + 4 * q^30 - 12 * q^31 - 12 * q^32 + 16 * q^37 - 4 * q^40 - 12 * q^41 - 8 * q^42 + 32 * q^43 + 24 * q^44 - 4 * q^45 + 28 * q^46 - 8 * q^50 + 24 * q^53 - 4 * q^56 - 16 * q^57 + 8 * q^58 - 4 * q^60 - 8 * q^61 + 12 * q^62 + 4 * q^63 + 12 * q^64 + 16 * q^65 + 12 * q^71 + 12 * q^73 - 16 * q^74 + 16 * q^77 - 20 * q^79 + 4 * q^80 - 12 * q^81 + 12 * q^82 + 16 * q^83 + 8 * q^84 + 60 * q^85 - 32 * q^86 - 24 * q^87 - 24 * q^88 - 72 * q^89 + 4 * q^90 - 24 * q^91 - 28 * q^92 + 16 * q^93 + 16 * q^95 - 20 * q^97 - 24 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 2x^{10} - 20x^{9} + 2x^{8} + 76x^{7} + 74x^{6} + 116x^{5} + 304x^{4} + 276x^{3} + 104x^{2} + 16x + 2 $ x^12 + 2*x^10 - 20*x^9 + 2*x^8 + 76*x^7 + 74*x^6 + 116*x^5 + 304*x^4 + 276*x^3 + 104*x^2 + 16*x + 2 :

$\beta_{1}$	$=$	$ ( 319660 \nu^{11} + 6162497 \nu^{10} - 2899224 \nu^{9} + 7687604 \nu^{8} - 131010082 \nu^{7} + \cdots + 86695495 ) / 92663677 $ (319660v^11 + 6162497v^10 - 2899224v^9 + 7687604v^8 - 131010082v^7 + 111983191v^6 + 446056166v^5 + 259986450v^4 + 655833134v^3 + 1613218458v^2 + 790288852*v + 86695495) / 92663677
$\beta_{2}$	$=$	$ ( - 12923481596 \nu^{11} - 57666526924 \nu^{10} + 38285706527 \nu^{9} + 103424348433 \nu^{8} + \cdots + 1036469068743 ) / 669495066325 $ (-12923481596v^11 - 57666526924v^10 + 38285706527v^9 + 103424348433v^8 + 1271142078210v^7 - 2461186861706v^6 - 4390093986768v^5 - 1242599579403v^4 - 8899767678291v^3 - 15161517707600v^2 - 2431186052684*v + 1036469068743) / 669495066325
$\beta_{3}$	$=$	$ ( 18908558264 \nu^{11} - 6443825434 \nu^{10} + 51879319657 \nu^{9} - 402143663647 \nu^{8} + \cdots + 778434883963 ) / 669495066325 $ (18908558264v^11 - 6443825434v^10 + 51879319657v^9 - 402143663647v^8 + 200233357185v^7 + 1116804730129v^6 + 1178357309887v^5 + 2637268036652v^4 + 5313752502419v^3 + 4186696256800v^2 + 3502095461806*v + 778434883963) / 669495066325
$\beta_{4}$	$=$	$ ( 15733124837 \nu^{11} + 14036152383 \nu^{10} + 18876105726 \nu^{9} - 280238400526 \nu^{8} + \cdots + 72744335919 ) / 133899013265 $ (15733124837v^11 + 14036152383v^10 + 18876105726v^9 - 280238400526v^8 - 276780662900v^7 + 1487783945432v^6 + 2080340430251v^5 + 1958912577291v^4 + 5993996785382v^3 + 7294672351030v^2 + 2312609131848*v + 72744335919) / 133899013265
$\beta_{5}$	$=$	$ ( - 21029498283 \nu^{11} + 3000308893 \nu^{10} - 48511924779 \nu^{9} + 434028626134 \nu^{8} + \cdots - 263276433211 ) / 133899013265 $ (-21029498283v^11 + 3000308893v^10 - 48511924779v^9 + 434028626134v^8 - 120070192260v^7 - 1446659043998v^6 - 1499595564244v^5 - 2586461889239v^4 - 5986001947163v^3 - 5495552813010v^2 - 2631681056162*v - 263276433211) / 133899013265
$\beta_{6}$	$=$	$ ( - 137602532648 \nu^{11} + 39009615588 \nu^{10} - 311897131724 \nu^{9} + 2866433521479 \nu^{8} + \cdots - 212131664041 ) / 669495066325 $ (-137602532648v^11 + 39009615588v^10 - 311897131724v^9 + 2866433521479v^8 - 1162729657420v^7 - 9553493390053v^6 - 8091302379609v^5 - 15080586221489v^4 - 37765626218808v^3 - 30217739331950v^2 - 11137375872242*v - 212131664041) / 669495066325
$\beta_{7}$	$=$	$ ( 152931165873 \nu^{11} - 44919774063 \nu^{10} + 322252025174 \nu^{9} - 3155867646579 \nu^{8} + \cdots + 489934648991 ) / 669495066325 $ (152931165873v^11 - 44919774063v^10 + 322252025174v^9 - 3155867646579v^8 + 1241880145545v^7 + 11180402250978v^6 + 8100779376059v^5 + 15524458094964v^4 + 42139978823283v^3 + 30091698619600v^2 + 7442564324742*v + 489934648991) / 669495066325
$\beta_{8}$	$=$	$ ( 156974051961 \nu^{11} - 129561015766 \nu^{10} + 366195750868 \nu^{9} - 3414483670903 \nu^{8} + \cdots - 1378068766463 ) / 669495066325 $ (156974051961v^11 - 129561015766v^10 + 366195750868v^9 - 3414483670903v^8 + 3006160564540v^7 + 10618838028896v^6 + 2153459601513v^5 + 12713326529623v^4 + 34732178246456v^3 + 9169042821700v^2 - 4621260405106*v - 1378068766463) / 669495066325
$\beta_{9}$	$=$	$ ( - 34892568606 \nu^{11} + 5558673211 \nu^{10} - 69443493368 \nu^{9} + 705503715068 \nu^{8} + \cdots - 501644724487 ) / 133899013265 $ (-34892568606v^11 + 5558673211v^10 - 69443493368v^9 + 705503715068v^8 - 175617446830v^7 - 2660007584581v^6 - 2077482558963v^5 - 3721196041423v^4 - 10078002716166v^3 - 7899141078690v^2 - 2509574945664*v - 501644724487) / 133899013265
$\beta_{10}$	$=$	$ ( 54747762541 \nu^{11} - 25525851741 \nu^{10} + 118845810378 \nu^{9} - 1148241361088 \nu^{8} + \cdots - 105960876263 ) / 133899013265 $ (54747762541v^11 - 25525851741v^10 + 118845810378v^9 - 1148241361088v^8 + 635874442040v^7 + 3922613587591v^6 + 2163512470518v^5 + 5239908066713v^4 + 14105062825121v^3 + 8183672494390v^2 + 1219655783724*v - 105960876263) / 133899013265
$\beta_{11}$	$=$	$ ( - 282869968078 \nu^{11} + 47147159718 \nu^{10} - 567735683114 \nu^{9} + 5732219292469 \nu^{8} + \cdots - 2333282292101 ) / 669495066325 $ (-282869968078v^11 + 47147159718v^10 - 567735683114v^9 + 5732219292469v^8 - 1486414130720v^7 - 21428590593533v^6 - 16877697115624v^5 - 30094145132929v^4 - 81511923051063v^3 - 64117759581350v^2 - 18731521731362*v - 2333282292101) / 669495066325

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

$\nu$	$=$	$ ( \beta_{11} - \beta_{9} + \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 2 $ (b11 - b9 + b7 - b4 + b3 - b2 + b1 - 1) / 2
$\nu^{2}$	$=$	$ ( \beta_{11} + \beta_{10} - 2\beta_{9} - 2\beta_{8} - \beta_{5} - 6\beta_{3} + \beta_{2} - 2\beta _1 - 2 ) / 2 $ (b11 + b10 - 2b9 - 2b8 - b5 - 6b3 + b2 - 2b1 - 2) / 2
$\nu^{3}$	$=$	$ ( - \beta_{11} + 7 \beta_{10} + \beta_{9} - 5 \beta_{8} - 12 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} + \cdots + 12 ) / 2 $ (-b11 + 7b10 + b9 - 5b8 - 12b7 - 7b6 + 5b5 + b4 + 4b3 - b2 - 4*b1 + 12) / 2
$\nu^{4}$	$=$	$ ( 17 \beta_{11} - 5 \beta_{10} - 9 \beta_{9} + 11 \beta_{8} + 24 \beta_{7} - 9 \beta_{6} + 17 \beta_{5} + \cdots + 46 \beta_1 ) / 2 $ (17b11 - 5b10 - 9b9 + 11b8 + 24b7 - 9b6 + 17b5 - 11b4 + 24b3 - 5b2 + 46*b1) / 2
$\nu^{5}$	$=$	$ ( 35 \beta_{11} - 11 \beta_{10} - 57 \beta_{9} - 11 \beta_{8} + 36 \beta_{7} + 17 \beta_{6} - 17 \beta_{5} + \cdots - 116 ) / 2 $ (35b11 - 11b10 - 57b9 - 11b8 + 36b7 + 17b6 - 17b5 + 35b4 - 116b3 + 57b2 - 36*b1 - 116) / 2
$\nu^{6}$	$=$	$ - 56 \beta_{11} + 56 \beta_{10} + 33 \beta_{9} - 33 \beta_{8} - 204 \beta_{7} - 70 \beta_{6} + \cdots + 120 $ -56b11 + 56b10 + 33b9 - 33b8 - 204b7 - 70b6 + 33b5 + 70b4 + 33b2 - 120b1 + 120
$\nu^{7}$	$=$	$ 56 \beta_{11} - 139 \beta_{10} + 56 \beta_{9} + 242 \beta_{8} + 181 \beta_{7} - 139 \beta_{6} + \cdots + 181 $ 56b11 - 139b10 + 56b9 + 242b8 + 181b7 - 139b6 + 242b5 - 110b4 + 544b3 - 110b2 + 544*b1 + 181
$\nu^{8}$	$=$	$ 358 \beta_{11} - 587 \beta_{10} - 553 \beta_{9} + 235 \beta_{8} + 1165 \beta_{7} + 553 \beta_{6} + \cdots - 1868 $ 358b11 - 587b10 - 553b9 + 235b8 + 1165b7 + 553b6 - 358b5 + 235b4 - 1165b3 + 587b2 - 1868
$\nu^{9}$	$=$	$ - 2108 \beta_{11} + 1248 \beta_{10} + 1163 \beta_{9} - 1248 \beta_{8} - 5089 \beta_{7} - 553 \beta_{6} + \cdots + 1805 $ -2108b11 + 1248b10 + 1163b9 - 1248b8 - 5089b7 - 553b6 - 553b5 + 2108b4 - 1805b3 + 1163b2 - 5089*b1 + 1805
$\nu^{10}$	$=$	$ - 1682 \beta_{11} - 1682 \beta_{10} + 5038 \beta_{9} + 5038 \beta_{8} - 3657 \beta_{6} + 5373 \beta_{5} + \cdots + 11191 $ -1682b11 - 1682b10 + 5038b9 + 5038b8 - 3657b6 + 5373b5 - 3657b4 + 17288b3 - 5373b2 + 11191b1 + 11191
$\nu^{11}$	$=$	$ 13200 \beta_{11} - 18720 \beta_{10} - 13200 \beta_{9} + 10025 \beta_{8} + 47670 \beta_{7} + 18720 \beta_{6} + \cdots - 47670 $ 13200b11 - 18720b10 - 13200b9 + 10025b8 + 47670b7 + 18720b6 - 10025b5 - 5373b4 - 17708b3 + 5373b2 + 17708*b1 - 47670

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/510\mathbb{Z}\right)^\times$.

$n$	$241$	$307$	$341$
$\chi(n)$	$\beta_{1}$	$-1$	$1$

Embeddings

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

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

259.1

−1.15599	+	0.478829i
−0.527307	+	0.218418i
2.39041	−	0.990140i
−0.0681564	−	0.164544i
0.535744	+	1.29340i
−1.17469	−	2.83596i
−1.15599	−	0.478829i
−0.527307	−	0.218418i
2.39041	+	0.990140i
−0.0681564	+	0.164544i
0.535744	−	1.29340i
−1.17469	+	2.83596i

−1.00000

−0.707107

−

0.707107i

1.00000

−1.29771

1.82097i

0.707107

0.707107i

−2.57524

2.57524i

−1.00000

1.00000i

1.29771

−

1.82097i

259.2

−1.00000

−0.707107

−

0.707107i

1.00000

0.836360

−

2.07377i

0.707107

0.707107i

2.93275

−

2.93275i

−1.00000

1.00000i

−0.836360

2.07377i

259.3

−1.00000

−0.707107

−

0.707107i

1.00000

2.16846

0.545687i

0.707107

0.707107i

−0.771718

0.771718i

−1.00000

1.00000i

−2.16846

−

0.545687i

259.4

−1.00000

0.707107

0.707107i

1.00000

−2.23520

−

0.0621645i

−0.707107

−

0.707107i

−0.0879139

0.0879139i

−1.00000

1.00000i

2.23520

0.0621645i

259.5

−1.00000

0.707107

0.707107i

1.00000

0.336029

2.21068i

−0.707107

−

0.707107i

3.12637

−

3.12637i

−1.00000

1.00000i

−0.336029

−

2.21068i

259.6

−1.00000

0.707107

0.707107i

1.00000

2.19207

−

0.441404i

−0.707107

−

0.707107i

−0.624239

0.624239i

−1.00000

1.00000i

−2.19207

0.441404i

319.1

−1.00000

−0.707107

0.707107i

1.00000

−1.29771

−

1.82097i

0.707107

−

0.707107i

−2.57524

−

2.57524i

−1.00000

−

1.00000i

1.29771

1.82097i

319.2

−1.00000

−0.707107

0.707107i

1.00000

0.836360

2.07377i

0.707107

−

0.707107i

2.93275

2.93275i

−1.00000

−

1.00000i

−0.836360

−

2.07377i

319.3

−1.00000

−0.707107

0.707107i

1.00000

2.16846

−

0.545687i

0.707107

−

0.707107i

−0.771718

−

0.771718i

−1.00000

−

1.00000i

−2.16846

0.545687i

319.4

−1.00000

0.707107

−

0.707107i

1.00000

−2.23520

0.0621645i

−0.707107

0.707107i

−0.0879139

−

0.0879139i

−1.00000

−

1.00000i

2.23520

−

0.0621645i

319.5

−1.00000

0.707107

−

0.707107i

1.00000

0.336029

−

2.21068i

−0.707107

0.707107i

3.12637

3.12637i

−1.00000

−

1.00000i

−0.336029

2.21068i

319.6

−1.00000

0.707107

−

0.707107i

1.00000

2.19207

0.441404i

−0.707107

0.707107i

−0.624239

−

0.624239i

−1.00000

−

1.00000i

−2.19207

−

0.441404i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
85.j	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	510.2.m.c	✓	12
3.b	odd	2	1		1530.2.n.t		12
5.b	even	2	1		510.2.m.d	yes	12
15.d	odd	2	1		1530.2.n.s		12
17.c	even	4	1		510.2.m.d	yes	12
51.f	odd	4	1		1530.2.n.s		12
85.j	even	4	1	inner	510.2.m.c	✓	12
255.i	odd	4	1		1530.2.n.t		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
510.2.m.c	✓	12	1.a	even	1	1	trivial
510.2.m.c	✓	12	85.j	even	4	1	inner
510.2.m.d	yes	12	5.b	even	2	1
510.2.m.d	yes	12	17.c	even	4	1
1530.2.n.s		12	15.d	odd	2	1
1530.2.n.s		12	51.f	odd	4	1
1530.2.n.t		12	3.b	odd	2	1
1530.2.n.t		12	255.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{12} - 4 T_{7}^{11} + 8 T_{7}^{10} + 40 T_{7}^{9} + 236 T_{7}^{8} - 672 T_{7}^{7} + 1600 T_{7}^{6} + \cdots + 64 $ T7^12 - 4*T7^11 + 8*T7^10 + 40*T7^9 + 236*T7^8 - 672*T7^7 + 1600*T7^6 + 8448*T7^5 + 15728*T7^4 + 13504*T7^3 + 6272*T7^2 + 896*T7 + 64 acting on $S_{2}^{\mathrm{new}}(510, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T + 1)^{12} $ (T + 1)^12
$3$	$ (T^{4} + 1)^{3} $ (T^4 + 1)^3
$5$	$ T^{12} - 4 T^{11} + \cdots + 15625 $ T^12 - 4T^11 + 4T^10 + 4T^9 - 37T^8 + 80T^7 - 72T^6 + 400T^5 - 925T^4 + 500T^3 + 2500T^2 - 12500*T + 15625
$7$	$ T^{12} - 4 T^{11} + \cdots + 64 $ T^12 - 4T^11 + 8T^10 + 40T^9 + 236T^8 - 672T^7 + 1600T^6 + 8448T^5 + 15728T^4 + 13504T^3 + 6272T^2 + 896*T + 64
$11$	$ (T^{2} - 4 T + 8)^{6} $ (T^2 - 4*T + 8)^6
$13$	$ T^{12} + 80 T^{10} + \cdots + 541696 $ T^12 + 80T^10 + 2400T^8 + 33600T^6 + 221952T^4 + 607232*T^2 + 541696
$17$	$ T^{12} + 6 T^{10} + \cdots + 24137569 $ T^12 + 6T^10 + 64T^9 + 239T^8 + 192T^7 + 5268T^6 + 3264T^5 + 69071T^4 + 314432T^3 + 501126*T^2 + 24137569
$19$	$ T^{12} + 160 T^{10} + \cdots + 16384 $ T^12 + 160T^10 + 8448T^8 + 164608T^6 + 905216T^4 + 1114112*T^2 + 16384
$23$	$ T^{12} + 28 T^{11} + \cdots + 5184 $ T^12 + 28T^11 + 392T^10 + 3336T^9 + 18876T^8 + 72320T^7 + 190016T^6 + 335168T^5 + 394288T^4 + 293952T^3 + 139392T^2 + 38016*T + 5184
$29$	$ T^{12} + 8 T^{11} + \cdots + 84640000 $ T^12 + 8T^11 + 32T^10 - 128T^9 + 1888T^8 + 13344T^7 + 54528T^6 - 223488T^5 + 620224T^4 + 1644800T^3 + 8000000T^2 - 36800000*T + 84640000
$31$	$ T^{12} + 12 T^{11} + \cdots + 254016 $ T^12 + 12T^11 + 72T^10 - 376T^9 + 3452T^8 + 37632T^7 + 273728T^6 - 1824448T^5 + 5310256T^4 - 5374656T^3 + 2332800T^2 + 1088640*T + 254016
$37$	$ T^{12} - 16 T^{11} + \cdots + 19927296 $ T^12 - 16T^11 + 128T^10 - 224T^9 + 224T^8 - 7136T^7 + 110592T^6 - 84864T^5 - 171584T^4 + 983808T^3 + 25920000T^2 + 32140800*T + 19927296
$41$	$ T^{12} + \cdots + 962985024 $ T^12 + 12T^11 + 72T^10 + 8T^9 + 13100T^8 + 162528T^7 + 1007168T^6 - 234304T^5 + 267952T^4 + 41802432T^3 + 494236800T^2 - 975646080*T + 962985024
$43$	$ (T^{6} - 16 T^{5} + \cdots + 2592)^{2} $ (T^6 - 16T^5 + 40T^4 + 432T^3 - 2336T^2 + 2304*T + 2592)^2
$47$	$ T^{12} + \cdots + 162205696 $ T^12 + 248T^10 + 21424T^8 + 819200T^6 + 14616320T^4 + 107661312*T^2 + 162205696
$53$	$ (T^{6} - 12 T^{5} + \cdots + 18496)^{2} $ (T^6 - 12T^5 - 52T^4 + 832T^3 + 80T^2 - 14144*T + 18496)^2
$59$	$ T^{12} + 176 T^{10} + \cdots + 16516096 $ T^12 + 176T^10 + 9888T^8 + 227776T^6 + 2360064T^4 + 10695680*T^2 + 16516096
$61$	$ T^{12} + \cdots + 888516864 $ T^12 + 8T^11 + 32T^10 - 656T^9 + 10736T^8 + 44128T^7 + 224640T^6 - 4998144T^5 + 45696064T^4 - 46425600T^3 + 7008768T^2 + 111601152*T + 888516864
$67$	$ T^{12} + 496 T^{10} + \cdots + 85525504 $ T^12 + 496T^10 + 87776T^8 + 6927296T^6 + 235057920T^4 + 2371927040*T^2 + 85525504
$71$	$ T^{12} + \cdots + 68346690624 $ T^12 - 12T^11 + 72T^10 + 952T^9 + 17900T^8 - 148320T^7 + 944192T^6 + 14083840T^5 + 130163056T^4 - 234088896T^3 + 943516800T^2 + 11356606080*T + 68346690624
$73$	$ T^{12} + \cdots + 277830193216 $ T^12 - 12T^11 + 72T^10 + 376T^9 + 72492T^8 - 823776T^7 + 4736576T^6 + 18313280T^5 + 1296290736T^4 - 14025180864T^3 + 77794612352T^2 + 207911963008*T + 277830193216
$79$	$ T^{12} + \cdots + 5091107904 $ T^12 + 20T^11 + 200T^10 - 872T^9 + 17596T^8 + 293248T^7 + 2725952T^6 - 10415680T^5 + 45707056T^4 + 361869504T^3 + 2209795200T^2 - 4743480960*T + 5091107904
$83$	$ (T^{6} - 8 T^{5} + \cdots - 1579904)^{2} $ (T^6 - 8T^5 - 368T^4 + 2208T^3 + 42880T^2 - 147712*T - 1579904)^2
$89$	$ (T^{6} + 36 T^{5} + \cdots - 138176)^{2} $ (T^6 + 36T^5 + 412T^4 + 896T^3 - 13520T^2 - 86592*T - 138176)^2
$97$	$ T^{12} + \cdots + 5484291136 $ T^12 + 20T^11 + 200T^10 + 440T^9 + 41836T^8 + 749216T^7 + 6713920T^6 + 28956480T^5 + 67014064T^4 + 71293248T^3 + 542653568T^2 + 2439700864*T + 5484291136
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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:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - q^{2} - \beta_{7} q^{3} + q^{4} - \beta_{2} q^{5} + \beta_{7} q^{6} + (\beta_{6} + \beta_{4}) q^{7} - q^{8} - \beta_1 q^{9}+O(q^{10}) \) q - q^2 - b7 * q^3 + q^4 - b2 * q^5 + b7 * q^6 + (b6 + b4) * q^7 - q^8 - b1 * q^9	\( q - q^{2} - \beta_{7} q^{3} + q^{4} - \beta_{2} q^{5} + \beta_{7} q^{6} + (\beta_{6} + \beta_{4}) q^{7} - q^{8} - \beta_1 q^{9} + \beta_{2} q^{10} + ( - 2 \beta_1 + 2) q^{11} - \beta_{7} q^{12} + ( - \beta_{11} + \beta_{9} + \cdots + 2 \beta_1) q^{13}+ \cdots + ( - 2 \beta_1 - 2) q^{99}+O(q^{100}) \) q - q^2 - b7 * q^3 + q^4 - b2 * q^5 + b7 * q^6 + (b6 + b4) * q^7 - q^8 - b1 * q^9 + b2 * q^10 + (-2b1 + 2) q^11 - b7 * q^12 + (-b11 + b9 - b7 + b6 - b5 - b3 + 2b1) q^13 + (-b6 - b4) * q^14 - b6 * q^15 + q^16 + (b10 + b8 - b7 + b4 - b3 + b2 - b1) * q^17 + b1 * q^18 + (b9 + b8 - 2b7 + b6 - b4 - 2b3 + 2b1) q^19 - b2 * q^20 + (b11 - b5) * q^21 + (2b1 - 2) q^22 + (b11 + b10 - 3b1 - 3) q^23 + b7 * q^24 + (-b11 + b9 - b8 - 3b7 - 2b4 + b3 - b2 + 2b1 + 1) q^25 + (b11 - b9 + b7 - b6 + b5 + b3 - 2b1) q^26 - b3 * q^27 + (b6 + b4) * q^28 + (b11 + b10 - b9 - b8 + b6 + b4 - 2b3 - 2b1 - 2) * q^29 + b6 * q^30 + (-2b11 - 2b10 + b9 + b8 - b6 + b5 - b4 + 2b3 - b2 + b1 + 1) q^31 - q^32 + (-2b7 - 2b3) * q^33 + (-b10 - b8 + b7 - b4 + b3 - b2 + b1) * q^34 + (-b10 + b8 - 4b7 + b6 - 2b5 - 2b3 - b2 + 3b1 - 1) * q^35 - b1 * q^36 + (-b9 + b8 + 2b7 + b6 - b5 - b4 - b2) q^37 + (-b9 - b8 + 2b7 - b6 + b4 + 2b3 - 2b1) q^38 + (b11 + b10 - b9 - b8 + 2b3 - b1 - 1) q^39 + b2 * q^40 + (-b11 + b10 + b9 - b8 - 4b7 - 2b5 - 2b2 + 2b1 - 2) * q^41 + (-b11 + b5) * q^42 + (b11 + b8 + 3b7 - b5 + b4 - 3b3 + 2) * q^43 + (-2b1 + 2) q^44 - b11 * q^45 + (-b11 - b10 + 3b1 + 3) q^46 + (b11 - b10 - 2b7 + b5 - 2b3 - b2 + 2b1) q^47 - b7 * q^48 + (-b11 + b10 + 3b9 - b8 - 4b7 + 3b6 - b5 + b4 - 4b3 + b2 + b1) * q^49 + (b11 - b9 + b8 + 3b7 + 2b4 - b3 + b2 - 2b1 - 1) q^50 + (b6 - b5 + b4 - b3 - b2 - b1 - 1) * q^51 + (-b11 + b9 - b7 + b6 - b5 - b3 + 2b1) q^52 + (-b11 + b10 - 2b8 - 2b7 + b5 - 2b4 + 2b3 - b2 + 2) * q^53 + b3 * q^54 + (-2b11 - 2b2) * q^55 + (-b6 - b4) * q^56 + (b11 + b10 + b5 + 2b3 - b2 - 2b1 - 2) * q^57 + (-b11 - b10 + b9 + b8 - b6 - b4 + 2b3 + 2b1 + 2) * q^58 + (b10 + b8 + b7 - b4 + b3 + b2) * q^59 - b6 * q^60 + (-b11 + b10 + b9 - b8 - 6b7 + b6 - 2b5 - b4 - 2b2 + 2b1 - 2) * q^61 + (2b11 + 2b10 - b9 - b8 + b6 - b5 + b4 - 2b3 + b2 - b1 - 1) q^62 + (-b9 + b8) * q^63 + q^64 + (b11 - b10 + 2b9 + b6 - 2b5 + b4 - 3b1 + 1) q^65 + (2b7 + 2b3) * q^66 + (2b11 + 3b10 - b8 + b7 + 2b5 + b4 + b3 + 3b2 - 4b1) q^67 + (b10 + b8 - b7 + b4 - b3 + b2 - b1) * q^68 + (b8 + 3b7 + b4 - 3b3) * q^69 + (b10 - b8 + 4b7 - b6 + 2b5 + 2b3 + b2 - 3b1 + 1) * q^70 + (-b11 - b10 - b9 - b8 - 2b6 - b5 - 2b4 - 2b3 + b2 + 2b1 + 2) * q^71 + b1 * q^72 + (2b11 - 2b10 - 2b7 + 3b6 - 3b5 - 3b4 - 3b2 + 2b1 - 2) * q^73 + (b9 - b8 - 2b7 - b6 + b5 + b4 + b2) q^74 + (b10 - b8 - b7 - b6 + 2b5 + 2b3 + b2 - 3b1 + 1) q^75 + (b9 + b8 - 2b7 + b6 - b4 - 2b3 + 2b1) q^76 + (-2b9 + 2b8 + 2b6 + 2b4) * q^77 + (-b11 - b10 + b9 + b8 - 2b3 + b1 + 1) q^78 + (b11 - b10 - 3b9 + 3b8 + 2b7 - b6 + 2b5 + b4 + 2b2 + b1 - 1) q^79 - b2 * q^80 - q^81 + (b11 - b10 - b9 + b8 + 4b7 + 2b5 + 2b2 - 2b1 + 2) * q^82 + (-2b11 - 2b10 + 3b9 - b8 - 2b7 - 3b6 + 2b5 - b4 + 2b3 + 2b2 + 6) * q^83 + (b11 - b5) * q^84 + (-b11 - 2b10 - b9 + b8 - 2b6 + b5 + 2b4 + b2 - b1 + 7) q^85 + (-b11 - b8 - 3b7 + b5 - b4 + 3b3 - 2) * q^86 + (b11 - b10 + b8 + 2b7 - b5 + b4 - 2b3 + b2 - 2) * q^87 + (2b1 - 2) q^88 + (b11 + b10 - b9 + b8 - 2b7 + b6 - b5 + b4 + 2b3 - b2 - 8) * q^89 + b11 * q^90 + (b11 - b10 + 3b9 - 3b8 + 2b7 - b6 + b5 + b4 + b2) q^91 + (b11 + b10 - 3b1 - 3) q^92 + (-b11 + b10 + b9 - 2b8 - b7 - b6 + b5 - 2b4 + b3 - b2 + 2) * q^93 + (-b11 + b10 + 2b7 - b5 + 2b3 + b2 - 2b1) q^94 + (b11 - 3b10 + 8b7 - 2b6 - b5 + 4b3 - b2 + 4b1 + 2) q^95 + b7 * q^96 + (-2b9 + 2b8 + 2b7 - b6 + 3b5 + b4 + 3b2) q^97 + (b11 - b10 - 3b9 + b8 + 4b7 - 3b6 + b5 - b4 + 4b3 - b2 - b1) * q^98 + (-2b1 - 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} + 4 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 + 4 * q^5 + 4 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} + 4 q^{5} + 4 q^{7} - 12 q^{8} - 4 q^{10} + 24 q^{11} - 4 q^{14} - 4 q^{15} + 12 q^{16} + 4 q^{20} + 8 q^{21} - 24 q^{22} - 28 q^{23} + 8 q^{25} + 4 q^{28} - 8 q^{29} + 4 q^{30} - 12 q^{31} - 12 q^{32} + 16 q^{37} - 4 q^{40} - 12 q^{41} - 8 q^{42} + 32 q^{43} + 24 q^{44} - 4 q^{45} + 28 q^{46} - 8 q^{50} + 24 q^{53} - 4 q^{56} - 16 q^{57} + 8 q^{58} - 4 q^{60} - 8 q^{61} + 12 q^{62} + 4 q^{63} + 12 q^{64} + 16 q^{65} + 12 q^{71} + 12 q^{73} - 16 q^{74} + 16 q^{77} - 20 q^{79} + 4 q^{80} - 12 q^{81} + 12 q^{82} + 16 q^{83} + 8 q^{84} + 60 q^{85} - 32 q^{86} - 24 q^{87} - 24 q^{88} - 72 q^{89} + 4 q^{90} - 24 q^{91} - 28 q^{92} + 16 q^{93} + 16 q^{95} - 20 q^{97} - 24 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 + 4 * q^5 + 4 * q^7 - 12 * q^8 - 4 * q^10 + 24 * q^11 - 4 * q^14 - 4 * q^15 + 12 * q^16 + 4 * q^20 + 8 * q^21 - 24 * q^22 - 28 * q^23 + 8 * q^25 + 4 * q^28 - 8 * q^29 + 4 * q^30 - 12 * q^31 - 12 * q^32 + 16 * q^37 - 4 * q^40 - 12 * q^41 - 8 * q^42 + 32 * q^43 + 24 * q^44 - 4 * q^45 + 28 * q^46 - 8 * q^50 + 24 * q^53 - 4 * q^56 - 16 * q^57 + 8 * q^58 - 4 * q^60 - 8 * q^61 + 12 * q^62 + 4 * q^63 + 12 * q^64 + 16 * q^65 + 12 * q^71 + 12 * q^73 - 16 * q^74 + 16 * q^77 - 20 * q^79 + 4 * q^80 - 12 * q^81 + 12 * q^82 + 16 * q^83 + 8 * q^84 + 60 * q^85 - 32 * q^86 - 24 * q^87 - 24 * q^88 - 72 * q^89 + 4 * q^90 - 24 * q^91 - 28 * q^92 + 16 * q^93 + 16 * q^95 - 20 * q^97 - 24 * 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	510.2.m.c

Level	$510$
Weight	$2$
Character orbit	510.m
Analytic conductor	$4.072$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(4.07237050309\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} + 2x^{10} - 20x^{9} + 2x^{8} + 76x^{7} + 74x^{6} + 116x^{5} + 304x^{4} + 276x^{3} + 104x^{2} + 16x + 2 \) x^12 + 2x^10 - 20x^9 + 2x^8 + 76x^7 + 74x^6 + 116x^5 + 304x^4 + 276x^3 + 104x^2 + 16x + 2
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 319660 \nu^{11} + 6162497 \nu^{10} - 2899224 \nu^{9} + 7687604 \nu^{8} - 131010082 \nu^{7} + \cdots + 86695495 ) / 92663677 \) (319660v^11 + 6162497v^10 - 2899224v^9 + 7687604v^8 - 131010082v^7 + 111983191v^6 + 446056166v^5 + 259986450v^4 + 655833134v^3 + 1613218458v^2 + 790288852*v + 86695495) / 92663677
\(\beta_{2}\)	\(=\)	\( ( - 12923481596 \nu^{11} - 57666526924 \nu^{10} + 38285706527 \nu^{9} + 103424348433 \nu^{8} + \cdots + 1036469068743 ) / 669495066325 \) (-12923481596v^11 - 57666526924v^10 + 38285706527v^9 + 103424348433v^8 + 1271142078210v^7 - 2461186861706v^6 - 4390093986768v^5 - 1242599579403v^4 - 8899767678291v^3 - 15161517707600v^2 - 2431186052684*v + 1036469068743) / 669495066325
\(\beta_{3}\)	\(=\)	\( ( 18908558264 \nu^{11} - 6443825434 \nu^{10} + 51879319657 \nu^{9} - 402143663647 \nu^{8} + \cdots + 778434883963 ) / 669495066325 \) (18908558264v^11 - 6443825434v^10 + 51879319657v^9 - 402143663647v^8 + 200233357185v^7 + 1116804730129v^6 + 1178357309887v^5 + 2637268036652v^4 + 5313752502419v^3 + 4186696256800v^2 + 3502095461806*v + 778434883963) / 669495066325
\(\beta_{4}\)	\(=\)	\( ( 15733124837 \nu^{11} + 14036152383 \nu^{10} + 18876105726 \nu^{9} - 280238400526 \nu^{8} + \cdots + 72744335919 ) / 133899013265 \) (15733124837v^11 + 14036152383v^10 + 18876105726v^9 - 280238400526v^8 - 276780662900v^7 + 1487783945432v^6 + 2080340430251v^5 + 1958912577291v^4 + 5993996785382v^3 + 7294672351030v^2 + 2312609131848*v + 72744335919) / 133899013265
\(\beta_{5}\)	\(=\)	\( ( - 21029498283 \nu^{11} + 3000308893 \nu^{10} - 48511924779 \nu^{9} + 434028626134 \nu^{8} + \cdots - 263276433211 ) / 133899013265 \) (-21029498283v^11 + 3000308893v^10 - 48511924779v^9 + 434028626134v^8 - 120070192260v^7 - 1446659043998v^6 - 1499595564244v^5 - 2586461889239v^4 - 5986001947163v^3 - 5495552813010v^2 - 2631681056162*v - 263276433211) / 133899013265
\(\beta_{6}\)	\(=\)	\( ( - 137602532648 \nu^{11} + 39009615588 \nu^{10} - 311897131724 \nu^{9} + 2866433521479 \nu^{8} + \cdots - 212131664041 ) / 669495066325 \) (-137602532648v^11 + 39009615588v^10 - 311897131724v^9 + 2866433521479v^8 - 1162729657420v^7 - 9553493390053v^6 - 8091302379609v^5 - 15080586221489v^4 - 37765626218808v^3 - 30217739331950v^2 - 11137375872242*v - 212131664041) / 669495066325
\(\beta_{7}\)	\(=\)	\( ( 152931165873 \nu^{11} - 44919774063 \nu^{10} + 322252025174 \nu^{9} - 3155867646579 \nu^{8} + \cdots + 489934648991 ) / 669495066325 \) (152931165873v^11 - 44919774063v^10 + 322252025174v^9 - 3155867646579v^8 + 1241880145545v^7 + 11180402250978v^6 + 8100779376059v^5 + 15524458094964v^4 + 42139978823283v^3 + 30091698619600v^2 + 7442564324742*v + 489934648991) / 669495066325
\(\beta_{8}\)	\(=\)	\( ( 156974051961 \nu^{11} - 129561015766 \nu^{10} + 366195750868 \nu^{9} - 3414483670903 \nu^{8} + \cdots - 1378068766463 ) / 669495066325 \) (156974051961v^11 - 129561015766v^10 + 366195750868v^9 - 3414483670903v^8 + 3006160564540v^7 + 10618838028896v^6 + 2153459601513v^5 + 12713326529623v^4 + 34732178246456v^3 + 9169042821700v^2 - 4621260405106*v - 1378068766463) / 669495066325
\(\beta_{9}\)	\(=\)	\( ( - 34892568606 \nu^{11} + 5558673211 \nu^{10} - 69443493368 \nu^{9} + 705503715068 \nu^{8} + \cdots - 501644724487 ) / 133899013265 \) (-34892568606v^11 + 5558673211v^10 - 69443493368v^9 + 705503715068v^8 - 175617446830v^7 - 2660007584581v^6 - 2077482558963v^5 - 3721196041423v^4 - 10078002716166v^3 - 7899141078690v^2 - 2509574945664*v - 501644724487) / 133899013265
\(\beta_{10}\)	\(=\)	\( ( 54747762541 \nu^{11} - 25525851741 \nu^{10} + 118845810378 \nu^{9} - 1148241361088 \nu^{8} + \cdots - 105960876263 ) / 133899013265 \) (54747762541v^11 - 25525851741v^10 + 118845810378v^9 - 1148241361088v^8 + 635874442040v^7 + 3922613587591v^6 + 2163512470518v^5 + 5239908066713v^4 + 14105062825121v^3 + 8183672494390v^2 + 1219655783724*v - 105960876263) / 133899013265
\(\beta_{11}\)	\(=\)	\( ( - 282869968078 \nu^{11} + 47147159718 \nu^{10} - 567735683114 \nu^{9} + 5732219292469 \nu^{8} + \cdots - 2333282292101 ) / 669495066325 \) (-282869968078v^11 + 47147159718v^10 - 567735683114v^9 + 5732219292469v^8 - 1486414130720v^7 - 21428590593533v^6 - 16877697115624v^5 - 30094145132929v^4 - 81511923051063v^3 - 64117759581350v^2 - 18731521731362*v - 2333282292101) / 669495066325

\(\nu\)	\(=\)	\( ( \beta_{11} - \beta_{9} + \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 2 \) (b11 - b9 + b7 - b4 + b3 - b2 + b1 - 1) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} + \beta_{10} - 2\beta_{9} - 2\beta_{8} - \beta_{5} - 6\beta_{3} + \beta_{2} - 2\beta _1 - 2 ) / 2 \) (b11 + b10 - 2b9 - 2b8 - b5 - 6b3 + b2 - 2b1 - 2) / 2
\(\nu^{3}\)	\(=\)	\( ( - \beta_{11} + 7 \beta_{10} + \beta_{9} - 5 \beta_{8} - 12 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} + \cdots + 12 ) / 2 \) (-b11 + 7b10 + b9 - 5b8 - 12b7 - 7b6 + 5b5 + b4 + 4b3 - b2 - 4*b1 + 12) / 2
\(\nu^{4}\)	\(=\)	\( ( 17 \beta_{11} - 5 \beta_{10} - 9 \beta_{9} + 11 \beta_{8} + 24 \beta_{7} - 9 \beta_{6} + 17 \beta_{5} + \cdots + 46 \beta_1 ) / 2 \) (17b11 - 5b10 - 9b9 + 11b8 + 24b7 - 9b6 + 17b5 - 11b4 + 24b3 - 5b2 + 46*b1) / 2
\(\nu^{5}\)	\(=\)	\( ( 35 \beta_{11} - 11 \beta_{10} - 57 \beta_{9} - 11 \beta_{8} + 36 \beta_{7} + 17 \beta_{6} - 17 \beta_{5} + \cdots - 116 ) / 2 \) (35b11 - 11b10 - 57b9 - 11b8 + 36b7 + 17b6 - 17b5 + 35b4 - 116b3 + 57b2 - 36*b1 - 116) / 2
\(\nu^{6}\)	\(=\)	\( - 56 \beta_{11} + 56 \beta_{10} + 33 \beta_{9} - 33 \beta_{8} - 204 \beta_{7} - 70 \beta_{6} + \cdots + 120 \) -56b11 + 56b10 + 33b9 - 33b8 - 204b7 - 70b6 + 33b5 + 70b4 + 33b2 - 120b1 + 120
\(\nu^{7}\)	\(=\)	\( 56 \beta_{11} - 139 \beta_{10} + 56 \beta_{9} + 242 \beta_{8} + 181 \beta_{7} - 139 \beta_{6} + \cdots + 181 \) 56b11 - 139b10 + 56b9 + 242b8 + 181b7 - 139b6 + 242b5 - 110b4 + 544b3 - 110b2 + 544*b1 + 181
\(\nu^{8}\)	\(=\)	\( 358 \beta_{11} - 587 \beta_{10} - 553 \beta_{9} + 235 \beta_{8} + 1165 \beta_{7} + 553 \beta_{6} + \cdots - 1868 \) 358b11 - 587b10 - 553b9 + 235b8 + 1165b7 + 553b6 - 358b5 + 235b4 - 1165b3 + 587b2 - 1868
\(\nu^{9}\)	\(=\)	\( - 2108 \beta_{11} + 1248 \beta_{10} + 1163 \beta_{9} - 1248 \beta_{8} - 5089 \beta_{7} - 553 \beta_{6} + \cdots + 1805 \) -2108b11 + 1248b10 + 1163b9 - 1248b8 - 5089b7 - 553b6 - 553b5 + 2108b4 - 1805b3 + 1163b2 - 5089*b1 + 1805
\(\nu^{10}\)	\(=\)	\( - 1682 \beta_{11} - 1682 \beta_{10} + 5038 \beta_{9} + 5038 \beta_{8} - 3657 \beta_{6} + 5373 \beta_{5} + \cdots + 11191 \) -1682b11 - 1682b10 + 5038b9 + 5038b8 - 3657b6 + 5373b5 - 3657b4 + 17288b3 - 5373b2 + 11191b1 + 11191
\(\nu^{11}\)	\(=\)	\( 13200 \beta_{11} - 18720 \beta_{10} - 13200 \beta_{9} + 10025 \beta_{8} + 47670 \beta_{7} + 18720 \beta_{6} + \cdots - 47670 \) 13200b11 - 18720b10 - 13200b9 + 10025b8 + 47670b7 + 18720b6 - 10025b5 - 5373b4 - 17708b3 + 5373b2 + 17708*b1 - 47670

\(n\)	\(241\)	\(307\)	\(341\)
\(\chi(n)\)	\(\beta_{1}\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 259.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T + 1)^{12} \) (T + 1)^12
$3$	\( (T^{4} + 1)^{3} \) (T^4 + 1)^3
$5$	\( T^{12} - 4 T^{11} + \cdots + 15625 \) T^12 - 4T^11 + 4T^10 + 4T^9 - 37T^8 + 80T^7 - 72T^6 + 400T^5 - 925T^4 + 500T^3 + 2500T^2 - 12500*T + 15625
$7$	\( T^{12} - 4 T^{11} + \cdots + 64 \) T^12 - 4T^11 + 8T^10 + 40T^9 + 236T^8 - 672T^7 + 1600T^6 + 8448T^5 + 15728T^4 + 13504T^3 + 6272T^2 + 896*T + 64
$11$	\( (T^{2} - 4 T + 8)^{6} \) (T^2 - 4*T + 8)^6
$13$	\( T^{12} + 80 T^{10} + \cdots + 541696 \) T^12 + 80T^10 + 2400T^8 + 33600T^6 + 221952T^4 + 607232*T^2 + 541696
$17$	\( T^{12} + 6 T^{10} + \cdots + 24137569 \) T^12 + 6T^10 + 64T^9 + 239T^8 + 192T^7 + 5268T^6 + 3264T^5 + 69071T^4 + 314432T^3 + 501126*T^2 + 24137569
$19$	\( T^{12} + 160 T^{10} + \cdots + 16384 \) T^12 + 160T^10 + 8448T^8 + 164608T^6 + 905216T^4 + 1114112*T^2 + 16384
$23$	\( T^{12} + 28 T^{11} + \cdots + 5184 \) T^12 + 28T^11 + 392T^10 + 3336T^9 + 18876T^8 + 72320T^7 + 190016T^6 + 335168T^5 + 394288T^4 + 293952T^3 + 139392T^2 + 38016*T + 5184
$29$	\( T^{12} + 8 T^{11} + \cdots + 84640000 \) T^12 + 8T^11 + 32T^10 - 128T^9 + 1888T^8 + 13344T^7 + 54528T^6 - 223488T^5 + 620224T^4 + 1644800T^3 + 8000000T^2 - 36800000*T + 84640000
$31$	\( T^{12} + 12 T^{11} + \cdots + 254016 \) T^12 + 12T^11 + 72T^10 - 376T^9 + 3452T^8 + 37632T^7 + 273728T^6 - 1824448T^5 + 5310256T^4 - 5374656T^3 + 2332800T^2 + 1088640*T + 254016
$37$	\( T^{12} - 16 T^{11} + \cdots + 19927296 \) T^12 - 16T^11 + 128T^10 - 224T^9 + 224T^8 - 7136T^7 + 110592T^6 - 84864T^5 - 171584T^4 + 983808T^3 + 25920000T^2 + 32140800*T + 19927296
$41$	\( T^{12} + \cdots + 962985024 \) T^12 + 12T^11 + 72T^10 + 8T^9 + 13100T^8 + 162528T^7 + 1007168T^6 - 234304T^5 + 267952T^4 + 41802432T^3 + 494236800T^2 - 975646080*T + 962985024
$43$	\( (T^{6} - 16 T^{5} + \cdots + 2592)^{2} \) (T^6 - 16T^5 + 40T^4 + 432T^3 - 2336T^2 + 2304*T + 2592)^2
$47$	\( T^{12} + \cdots + 162205696 \) T^12 + 248T^10 + 21424T^8 + 819200T^6 + 14616320T^4 + 107661312*T^2 + 162205696
$53$	\( (T^{6} - 12 T^{5} + \cdots + 18496)^{2} \) (T^6 - 12T^5 - 52T^4 + 832T^3 + 80T^2 - 14144*T + 18496)^2
$59$	\( T^{12} + 176 T^{10} + \cdots + 16516096 \) T^12 + 176T^10 + 9888T^8 + 227776T^6 + 2360064T^4 + 10695680*T^2 + 16516096
$61$	\( T^{12} + \cdots + 888516864 \) T^12 + 8T^11 + 32T^10 - 656T^9 + 10736T^8 + 44128T^7 + 224640T^6 - 4998144T^5 + 45696064T^4 - 46425600T^3 + 7008768T^2 + 111601152*T + 888516864
$67$	\( T^{12} + 496 T^{10} + \cdots + 85525504 \) T^12 + 496T^10 + 87776T^8 + 6927296T^6 + 235057920T^4 + 2371927040*T^2 + 85525504
$71$	\( T^{12} + \cdots + 68346690624 \) T^12 - 12T^11 + 72T^10 + 952T^9 + 17900T^8 - 148320T^7 + 944192T^6 + 14083840T^5 + 130163056T^4 - 234088896T^3 + 943516800T^2 + 11356606080*T + 68346690624
$73$	\( T^{12} + \cdots + 277830193216 \) T^12 - 12T^11 + 72T^10 + 376T^9 + 72492T^8 - 823776T^7 + 4736576T^6 + 18313280T^5 + 1296290736T^4 - 14025180864T^3 + 77794612352T^2 + 207911963008*T + 277830193216
$79$	\( T^{12} + \cdots + 5091107904 \) T^12 + 20T^11 + 200T^10 - 872T^9 + 17596T^8 + 293248T^7 + 2725952T^6 - 10415680T^5 + 45707056T^4 + 361869504T^3 + 2209795200T^2 - 4743480960*T + 5091107904
$83$	\( (T^{6} - 8 T^{5} + \cdots - 1579904)^{2} \) (T^6 - 8T^5 - 368T^4 + 2208T^3 + 42880T^2 - 147712*T - 1579904)^2
$89$	\( (T^{6} + 36 T^{5} + \cdots - 138176)^{2} \) (T^6 + 36T^5 + 412T^4 + 896T^3 - 13520T^2 - 86592*T - 138176)^2
$97$	\( T^{12} + \cdots + 5484291136 \) T^12 + 20T^11 + 200T^10 + 440T^9 + 41836T^8 + 749216T^7 + 6713920T^6 + 28956480T^5 + 67014064T^4 + 71293248T^3 + 542653568T^2 + 2439700864*T + 5484291136
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