LMFDB - Newform orbit 930.2.n.d

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} - 2 x^{11} + 13 x^{10} - 9 x^{9} + 60 x^{8} + x^{7} + 263 x^{6} + 823 x^{5} + 1931 x^{4} + \cdots + 400 $ x^12 - 2x^11 + 13x^10 - 9x^9 + 60x^8 + x^7 + 263x^6 + 823x^5 + 1931x^4 + 1170x^3 + 1340x^2 + 1000x + 400
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_{6} + \beta_{5} - \beta_{3} - 1) q^{2} + \beta_{5} q^{3} + \beta_{3} q^{4} - q^{5} - q^{6} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{7} + \beta_{6} q^{8} + \beta_{6} q^{9}+O(q^{10}) $ q + (-b6 + b5 - b3 - 1) * q^2 + b5 * q^3 + b3 * q^4 - q^5 - q^6 + (-b7 - b4 - b1) * q^7 + b6 * q^8 + b6 * q^9	$ q + ( - \beta_{6} + \beta_{5} - \beta_{3} - 1) q^{2} + \beta_{5} q^{3} + \beta_{3} q^{4} - q^{5} - q^{6} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{7} + \beta_{6} q^{8} + \beta_{6} q^{9} + (\beta_{6} - \beta_{5} + \beta_{3} + 1) q^{10} + (\beta_{8} - \beta_{5} + \beta_{3} + 1) q^{11} + (\beta_{6} - \beta_{5} + \beta_{3} + 1) q^{12} + (\beta_{11} + \beta_{10} - \beta_{9} + \cdots - 1) q^{13}+ \cdots + ( - \beta_{9} + \beta_{6} + \beta_{3} + 1) q^{99}+O(q^{100}) $ q + (-b6 + b5 - b3 - 1) * q^2 + b5 * q^3 + b3 * q^4 - q^5 - q^6 + (-b7 - b4 - b1) * q^7 + b6 * q^8 + b6 * q^9 + (b6 - b5 + b3 + 1) * q^10 + (b8 - b5 + b3 + 1) * q^11 + (b6 - b5 + b3 + 1) * q^12 + (b11 + b10 - b9 + b8 + 2b7 - b6 - b5 + 2b4 + 2b2 + 2b1 - 1) * q^13 + (b7 + b2) * q^14 - b5 * q^15 - b5 * q^16 + (-b11 - b10 + b9 + b7 - 2b5 + 2b3 + b2) * q^17 - b5 * q^18 + (-b10 + b7 - b6 + b5 - b3 - b2 - b1 - 1) * q^19 - b3 * q^20 + (b7 + b2 + b1) * q^21 + (b11 + b5 - b3) * q^22 + (-b11 - 2b10 + 2b9 - b7 - b6 + b5 - b4 - b3 - b2) * q^23 - b3 * q^24 + q^25 + (b9 + b6 + b3 - 2b2 + 2) q^26 - b3 * q^27 + (b7 + b1) * q^28 + (-b10 + b9 - b8 - b7 + b2 + b1) * q^29 + q^30 + (b11 + b10 - 2b9 + 2b8 - b7 + b5 - b2 + 1) * q^31 + q^32 + (-b10 + b3 + 1) * q^33 + (b11 - b9 + b7 + 2b6 + b1 + 2) q^34 + (b7 + b4 + b1) * q^35 + q^36 + (3b6 + 3b3 - 2) * q^37 + (-b8 - b7 - b4 + b3 + b1) * q^38 + (b11 - b6 - b5 + 2b4 + b3) q^39 - b6 * q^40 + (-b10 - b9 + b8 - b7 + b6 - b5) * q^41 + (b7 + b4 + b1) * q^42 + (-b10 - b9 + b8 - b7 + 5b6 - 5b5 + 4b3 - b2 - b1 + 4) q^43 + (-b11 - b10 + b9 - b8 - b6 - 1) * q^44 - b6 * q^45 + (b11 - b10 - b9 - b8 - b6 + b5 + b4 + b2 - 1) * q^46 + (b10 + b8 + 3b7 - 2b6 - b5 + 2b4 + 2b2 + 3b1 - 2) q^47 - b6 * q^48 + (b10 + b8 + 3b7 + 3b6 + b4 + b2 + 3b1 + 3) q^49 + (-b6 + b5 - b3 - 1) * q^50 + (-b11 - b10 + b7 - 2b5 + b4 + 2b3 + b1 + 2) * q^51 + (-b10 - 2b7 - b6 + b5 - 2b3 - 2) * q^52 + (-2b11 - 2b10 + 2b9 + 3b7 + b4 + 3b2) q^53 - b6 * q^54 + (-b8 + b5 - b3 - 1) * q^55 + (b4 + b1) * q^56 + (-b9 - b4 - 2b2 - b1 - 1) q^57 + (-b11 - b10 - b8 + b7 + b4 - b1) * q^58 + (-b10 - b8 + b7 - 2b5 + b4 + b2 + b1) q^59 + (-b6 + b5 - b3 - 1) * q^60 + (b11 - 3b9 + b8 - b6 - 2b4 - b3 - b2 - 2b1 + 1) q^61 + (b11 + b10 + b9 - b7 - b6 + b5 - b3 - b1 - 2) * q^62 + (b4 + b1) * q^63 + (-b6 + b5 - b3 - 1) * q^64 + (-b11 - b10 + b9 - b8 - 2b7 + b6 + b5 - 2b4 - 2b2 - 2b1 + 1) * q^65 + (-b8 + b5 - b3 - 1) * q^66 + (-2b11 + 2b9 - 2b8 - b4 + 2b2 - b1 + 2) * q^67 + (-b11 + b9 - b8 - 2b6 + b4 - 2b3 + b1 - 2) * q^68 + (-2b11 - 2b10 - b8 - b7 + b5 - b4 - 1) * q^69 + (-b7 - b2) * q^70 + (2b11 + 2b10 - 2b9 + 3b7 + 2b6 - 2b5 + b4 + 2b3 + 3b2) * q^71 + (-b6 + b5 - b3 - 1) * q^72 + (b8 - b7 - 4b5 - b4 + 7b3 - 3b1 + 4) q^73 + (5b6 - 5b5 + 2b3 + 2) q^74 + b5 * q^75 + (-b11 + b7 + b6 + 2b4 + b2) q^76 + (b10 + b8 - 2b7 + b6 - b5 - 2b4 - 2b2 - 2b1 + 1) * q^77 + (-b11 - b10 + b9 - b8 - 2b7 + b6 + b5 - 2b4 - 2b2 - 2b1 + 1) * q^78 + (2b11 - 2b7 - 3b6 - b5 - b4 + b3 - 2b2) * q^79 + b5 * q^80 + (-b6 + b5 - b3 - 1) * q^81 + (b11 + b10 - b8 - b5 - b1 + 1) * q^82 + (-2b10 + b9 - b8 + 3b7 - 3b3 - 3) q^83 + (-b7 - b2) * q^84 + (b11 + b10 - b9 - b7 + 2b5 - 2b3 - b2) * q^85 + (b11 + b10 - b8 - b7 - b5 - b4 - 4b3 - b1 + 1) q^86 + (-b11 - b8 + b4 + 2b2 + b1) q^87 + (-b9 + b6 + b3 + 1) * q^88 + (b8 + 2b5 + b3 + 3b1 - 2) * q^89 + b5 * q^90 + (-b10 - b9 + b8 - 5b7 - 5b6 + 5b5 + 2b3 - 3b2 - 3b1 + 2) * q^91 + (-2b11 + b9 - 2b8 + b6 - b4 + b3 - b2 - b1) * q^92 + (2b11 - b9 + b8 - b7 + b6 + b5 - b4 - b1) q^93 + (b11 + b8 + 2b6 + b4 + 2b3 - 2b2 + b1 + 3) q^94 + (b10 - b7 + b6 - b5 + b3 + b2 + b1 + 1) * q^95 + b5 * q^96 + (3b11 + 3b10 + 2b8 + 3b5 - 3b3 + 2b1 - 3) * q^97 + (b11 + b8 - 3b6 + 2b4 - 3b3 - b2 + 2b1 - 3) * q^98 + (-b9 + b6 + b3 + 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	\( 12 q - 3 q^{2} + 3 q^{3} - 3 q^{4} - 12 q^{5} - 12 q^{6} - q^{7} - 3 q^{8} - 3 q^{9} + 3 q^{10} + 6 q^{11} + 3 q^{12} - 6 q^{13} - q^{14} - 3 q^{15} - 3 q^{16} - 13 q^{17} - 3 q^{18} - 10 q^{19} + 3 q^{20} + q^{21} + 6 q^{22} + 8 q^{23} + 3 q^{24} + 12 q^{25} + 14 q^{26} + 3 q^{27} - q^{28} + 7 q^{29} + 12 q^{30} + 16 q^{31} + 12 q^{32} + 9 q^{33} + 17 q^{34} + q^{35} + 12 q^{36} - 42 q^{37} + q^{39} + 3 q^{40} - 3 q^{41} + q^{42} + 5 q^{43} - 9 q^{44} + 3 q^{45} - 2 q^{46} - 16 q^{47} + 3 q^{48} + 28 q^{49} - 3 q^{50} + 13 q^{51} - 6 q^{52} - q^{53} + 3 q^{54} - 6 q^{55} + 4 q^{56} - 20 q^{57} - 3 q^{58} - 3 q^{59} - 3 q^{60} + 8 q^{61} - 14 q^{62} + 4 q^{63} - 3 q^{64} + 6 q^{65} - 6 q^{66} + 24 q^{67} - 8 q^{68} - 8 q^{69} + q^{70} - 19 q^{71} - 3 q^{72} + 10 q^{73} - 12 q^{74} + 3 q^{75} + 6 q^{78} + 3 q^{79} + 3 q^{80} - 3 q^{81} + 7 q^{82} - 36 q^{83} + q^{84} + 13 q^{85} + 20 q^{86} + 8 q^{87} + 6 q^{88} - 15 q^{89} + 3 q^{90} + 51 q^{91} - 12 q^{92} - q^{93} + 24 q^{94} + 10 q^{95} + 3 q^{96} - 14 q^{97} - 12 q^{98} + 6 q^{99}+O(q^{100}) \) 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 + 6 * q^11 + 3 * q^12 - 6 * q^13 - q^14 - 3 * q^15 - 3 * q^16 - 13 * q^17 - 3 * q^18 - 10 * q^19 + 3 * q^20 + q^21 + 6 * q^22 + 8 * q^23 + 3 * q^24 + 12 * q^25 + 14 * q^26 + 3 * q^27 - q^28 + 7 * q^29 + 12 * q^30 + 16 * q^31 + 12 * q^32 + 9 * q^33 + 17 * q^34 + q^35 + 12 * q^36 - 42 * q^37 + q^39 + 3 * q^40 - 3 * q^41 + q^42 + 5 * q^43 - 9 * q^44 + 3 * q^45 - 2 * q^46 - 16 * q^47 + 3 * q^48 + 28 * q^49 - 3 * q^50 + 13 * q^51 - 6 * q^52 - q^53 + 3 * q^54 - 6 * q^55 + 4 * q^56 - 20 * q^57 - 3 * q^58 - 3 * q^59 - 3 * q^60 + 8 * q^61 - 14 * q^62 + 4 * q^63 - 3 * q^64 + 6 * q^65 - 6 * q^66 + 24 * q^67 - 8 * q^68 - 8 * q^69 + q^70 - 19 * q^71 - 3 * q^72 + 10 * q^73 - 12 * q^74 + 3 * q^75 + 6 * q^78 + 3 * q^79 + 3 * q^80 - 3 * q^81 + 7 * q^82 - 36 * q^83 + q^84 + 13 * q^85 + 20 * q^86 + 8 * q^87 + 6 * q^88 - 15 * q^89 + 3 * q^90 + 51 * q^91 - 12 * q^92 - q^93 + 24 * q^94 + 10 * q^95 + 3 * q^96 - 14 * q^97 - 12 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 2 x^{11} + 13 x^{10} - 9 x^{9} + 60 x^{8} + x^{7} + 263 x^{6} + 823 x^{5} + 1931 x^{4} + \cdots + 400 $ x^12 - 2*x^11 + 13*x^10 - 9*x^9 + 60*x^8 + x^7 + 263*x^6 + 823*x^5 + 1931*x^4 + 1170*x^3 + 1340*x^2 + 1000*x + 400 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 17849017864 \nu^{11} + 17625053431 \nu^{10} - 178529954095 \nu^{9} + \cdots + 60463522673300 ) / 107519805038390 $ (-17849017864v^11 + 17625053431v^10 - 178529954095v^9 - 111496966669v^8 - 495858189935v^7 - 1902177059364v^6 + 216123116006v^5 - 22850390835465v^4 - 38781999340004v^3 - 33360822706280v^2 - 14600309174840*v + 60463522673300) / 107519805038390
$\beta_{3}$	$=$	$ ( 23361760035 \nu^{11} - 106837550849 \nu^{10} + 512829808605 \nu^{9} + \cdots - 15010929677480 ) / 107519805038390 $ (23361760035v^11 - 106837550849v^10 + 512829808605v^9 - 1139133730021v^8 + 2950126373569v^7 - 3751671881315v^6 + 10498141078366v^5 + 3863106684695v^4 + 20833220372194v^3 - 23111477672266v^2 + 148807987112405*v - 15010929677480) / 107519805038390
$\beta_{4}$	$=$	$ ( - 146654331477 \nu^{11} + 214983815436 \nu^{10} - 1782233600297 \nu^{9} + \cdots - 205909846055800 ) / 215039610076780 $ (-146654331477v^11 + 214983815436v^10 - 1782233600297v^9 + 626624859189v^8 - 9624980905700v^7 - 637596034947v^6 - 48748704399099v^5 - 127545198552457v^4 - 382232489020961v^3 - 302595918088550v^2 - 315224424729940*v - 205909846055800) / 215039610076780
$\beta_{5}$	$=$	$ ( 259180521843 \nu^{11} - 820941472449 \nu^{10} + 4037834562557 \nu^{9} + \cdots + 53049709415200 ) / 215039610076780 $ (259180521843v^11 - 820941472449v^10 + 4037834562557v^9 - 6329673984486v^8 + 19051303779369v^7 - 17907584046427v^6 + 72430523469356v^5 + 134261867661482v^4 + 278674930944664v^3 - 257444750971091v^2 + 212990793099520*v + 53049709415200) / 215039610076780
$\beta_{6}$	$=$	$ ( - 604635226733 \nu^{11} + 1137874382010 \nu^{10} - 7789757733805 \nu^{9} + \cdots - 663036463432360 ) / 430079220153560 $ (-604635226733v^11 + 1137874382010v^10 - 7789757733805v^9 + 4727597224217v^8 - 36724101470656v^7 - 2588067986473v^6 - 166627772868235v^5 - 496750299137235v^4 - 1258952186163283v^3 - 862551212637626v^2 - 943654494647340*v - 663036463432360) / 430079220153560
$\beta_{7}$	$=$	$ ( 302580428763 \nu^{11} - 668487778598 \nu^{10} + 3997049287899 \nu^{9} + \cdots + 103672208737200 ) / 215039610076780 $ (302580428763v^11 - 668487778598v^10 + 3997049287899v^9 - 3500472468789v^8 + 18166764568270v^7 - 4266046224647v^6 + 79043701815307v^5 + 221802656734169v^4 + 560685961527401v^3 + 134311106170100v^2 + 206130812427800*v + 103672208737200) / 215039610076780
$\beta_{8}$	$=$	$ ( 377873811249 \nu^{11} - 1246157058016 \nu^{10} + 6108799410656 \nu^{9} + \cdots - 228519186690980 ) / 107519805038390 $ (377873811249v^11 - 1246157058016v^10 + 6108799410656v^9 - 10496150942691v^8 + 30636118502149v^7 - 34700982692066v^6 + 115791065278799v^5 + 168010412077976v^4 + 381784730133684v^3 - 359469623547681v^2 + 87053037871525*v - 228519186690980) / 107519805038390
$\beta_{9}$	$=$	$ ( - 811947191771 \nu^{11} + 1070796378400 \nu^{10} - 8730178441851 \nu^{9} + \cdots - 797530905591380 ) / 215039610076780 $ (-811947191771v^11 + 1070796378400v^10 - 8730178441851v^9 - 1669598790561v^8 - 32876644690520v^7 - 47642147786361v^6 - 156547132073865v^5 - 848523389812191v^4 - 1801907179914651v^3 - 1493427761388610v^2 - 583577757140760*v - 797530905591380) / 215039610076780
$\beta_{10}$	$=$	$ ( 2196856365657 \nu^{11} - 6036163855492 \nu^{10} + 32819025256117 \nu^{9} + \cdots + 17\!\cdots\!20 ) / 430079220153560 $ (2196856365657v^11 - 6036163855492v^10 + 32819025256117v^9 - 42245615072839v^8 + 156459753762298v^7 - 91084996607083v^6 + 604977651733493v^5 + 1468271773696657v^4 + 3005207185931801v^3 + 578163383466308v^2 + 3110122517771760*v + 1778835879579520) / 430079220153560
$\beta_{11}$	$=$	$ ( - 2487542474027 \nu^{11} + 6234274597186 \nu^{10} - 33979177074555 \nu^{9} + \cdots - 15\!\cdots\!60 ) / 430079220153560 $ (-2487542474027v^11 + 6234274597186v^10 - 33979177074555v^9 + 35569652132971v^8 - 145106089008796v^7 + 42032352335873v^6 - 567823275403729v^5 - 1830786108310665v^4 - 3452889698168529v^3 - 226203310184226v^2 - 1085086519421180*v - 1579157825751160) / 430079220153560

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} - 4\beta_{5} + \beta _1 - 1 $ -b11 + b9 + b7 - b6 - 4*b5 + b1 - 1
$\nu^{3}$	$=$	$ \beta_{10} + \beta_{9} - \beta_{8} + 7\beta_{7} + 3\beta_{6} - 3\beta_{5} - \beta_{2} - \beta_1 $ b10 + b9 - b8 + 7b7 + 3b6 - 3*b5 - b2 - b1
$\nu^{4}$	$=$	$ 11 \beta_{11} + 9 \beta_{10} - 9 \beta_{9} + 12 \beta_{7} + 28 \beta_{6} + 11 \beta_{5} + \cdots + 12 \beta_{2} $ 11b11 + 9b10 - 9b9 + 12b7 + 28b6 + 11b5 + 4b4 - 11b3 + 12*b2
$\nu^{5}$	$=$	$ 13\beta_{11} - 27\beta_{9} + 13\beta_{8} + 43\beta_{6} + 14\beta_{4} + 43\beta_{3} + 71\beta_{2} + 14\beta _1 + 21 $ 13b11 - 27b9 + 13b8 + 43b6 + 14b4 + 43b3 + 71b2 + 14b1 + 21
$\nu^{6}$	$=$	$ - 84 \beta_{11} - 84 \beta_{10} + 28 \beta_{8} - 127 \beta_{7} - 114 \beta_{5} - 127 \beta_{4} + \cdots + 114 $ -84b11 - 84b10 + 28b8 - 127b7 - 114b5 - 127b4 + 342b3 - 52b1 + 114
$\nu^{7}$	$=$	$ - 314 \beta_{11} - 155 \beta_{10} + 314 \beta_{9} - 155 \beta_{8} - 491 \beta_{7} - 499 \beta_{6} + \cdots - 499 $ -314b11 - 155b10 + 314b9 - 155b8 - 491b7 - 499b6 + 109b5 - 665b4 - 665b2 - 491b1 - 499
$\nu^{8}$	$=$	$ 329 \beta_{10} + 824 \beta_{9} - 824 \beta_{8} - 267 \beta_{7} - 1983 \beta_{6} + 1983 \beta_{5} + \cdots - 3185 $ 329b10 + 824b9 - 824b8 - 267b7 - 1983b6 + 1983b5 - 3185b3 - 1304b2 - 1304*b1 - 3185
$\nu^{9}$	$=$	$ 3494 \beta_{11} + 1861 \beta_{10} - 1861 \beta_{9} + 2088 \beta_{7} - 93 \beta_{6} + \cdots + 2088 \beta_{2} $ 3494b11 + 1861b10 - 1861b9 + 2088b7 - 93b6 + 5429b5 + 6466b4 - 5429b3 + 2088*b2
$\nu^{10}$	$=$	$ 8327 \beta_{11} - 12048 \beta_{9} + 8327 \beta_{8} + 12957 \beta_{6} + 13301 \beta_{4} + 12957 \beta_{3} + \cdots + 30924 $ 8327b11 - 12048b9 + 8327b8 + 12957b6 + 13301b4 + 12957b3 + 12779b2 + 13301b1 + 30924
$\nu^{11}$	$=$	$ - 21106 \beta_{11} - 21106 \beta_{10} + 17022 \beta_{8} - 24483 \beta_{7} - 57656 \beta_{5} + \cdots + 57656 $ -21106b11 - 21106b10 + 17022b8 - 24483b7 - 57656b5 - 24483b4 + 65148b3 + 40117b1 + 57656

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

Character values

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

$n$	$187$	$311$	$871$
$\chi(n)$	$1$	$1$	$\beta_{6}$

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

−0.781164	+	2.40418i
0.270443	−	0.832338i
1.01072	−	3.11068i
2.30431	+	1.67418i
−0.479361	−	0.348276i
−1.32494	−	0.962628i
2.30431	−	1.67418i
−0.479361	+	0.348276i
−1.32494	+	0.962628i
−0.781164	−	2.40418i
0.270443	+	0.832338i
1.01072	+	3.11068i

−0.809017

0.587785i

0.809017

0.587785i

0.309017

−

0.951057i

−1.00000

−0.482786

1.48586i

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

0.167143

−

0.514413i

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

0.624660

−

1.92251i

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

−3.72844

−

2.70887i

−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

0.775623

0.563523i

−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

2.14380

1.55756i

−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

−3.72844

2.70887i

−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

0.775623

−

0.563523i

−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

2.14380

−

1.55756i

−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

−0.482786

−

1.48586i

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

0.167143

0.514413i

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

0.624660

1.92251i

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.d	✓	12
31.d	even	5	1	inner	930.2.n.d	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.n.d	✓	12	1.a	even	1	1	trivial
930.2.n.d	✓	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} - 3 T_{7}^{10} - 17 T_{7}^{9} + 190 T_{7}^{8} - 543 T_{7}^{7} + 1417 T_{7}^{6} + \cdots + 400 $ T7^12 + T7^11 - 3*T7^10 - 17*T7^9 + 190*T7^8 - 543*T7^7 + 1417*T7^6 - 2116*T7^5 + 3421*T7^4 - 3850*T7^3 + 2960*T7^2 - 1200*T7 + 400 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 + 400 $ T^12 + T^11 - 3T^10 - 17T^9 + 190T^8 - 543T^7 + 1417T^6 - 2116T^5 + 3421T^4 - 3850T^3 + 2960T^2 - 1200T + 400
$11$	$ T^{12} - 6 T^{11} + \cdots + 24025 $ T^12 - 6T^11 + 31T^10 - 71T^9 + 181T^8 - 175T^7 + 1765T^6 - 3260T^5 + 16015T^4 - 13300T^3 + 38200T^2 - 44175*T + 24025
$13$	$ T^{12} + 6 T^{11} + \cdots + 1079521 $ T^12 + 6T^11 + 36T^10 + 214T^9 + 1872T^8 + 4654T^7 + 41001T^6 + 274132T^5 + 1841889T^4 + 6280122T^3 + 14820553T^2 - 2098780*T + 1079521
$17$	$ T^{12} + 13 T^{11} + \cdots + 13830961 $ T^12 + 13T^11 + 131T^10 + 710T^9 + 2850T^8 + 6113T^7 + 24429T^6 + 58058T^5 + 444690T^4 + 915050T^3 + 2765786T^2 + 6813208*T + 13830961
$19$	$ T^{12} + 10 T^{11} + \cdots + 160000 $ T^12 + 10T^11 + 25T^10 - 110T^9 + 2315T^8 + 20450T^7 + 97125T^6 + 443550T^5 + 3576025T^4 + 4917500T^3 + 13006000T^2 - 840000*T + 160000
$23$	$ T^{12} + \cdots + 257442025 $ T^12 - 8T^11 + 99T^10 - 787T^9 + 6706T^8 - 22715T^7 + 149300T^6 - 600535T^5 + 9329740T^4 - 55699925T^3 + 179059575T^2 - 210109275*T + 257442025
$29$	$ T^{12} - 7 T^{11} + \cdots + 74149321 $ T^12 - 7T^11 + 28T^10 - 116T^9 + 2853T^8 - 16794T^7 + 151487T^6 - 925241T^5 + 5424493T^4 - 20556439T^3 + 59446413T^2 - 62498638*T + 74149321
$31$	$ T^{12} + \cdots + 887503681 $ T^12 - 16T^11 + 102T^10 - 357T^9 + 268T^8 + 14988T^7 - 142027T^6 + 464628T^5 + 257548T^4 - 10635387T^3 + 94199142T^2 - 458066416*T + 887503681
$37$	$ (T^{2} + 7 T + 1)^{6} $ (T^2 + 7*T + 1)^6
$41$	$ T^{12} + 3 T^{11} + \cdots + 144400 $ T^12 + 3T^11 - 21T^10 - 133T^9 + 3436T^8 + 5305T^7 + 63515T^6 - 13030T^5 + 233065T^4 - 251450T^3 + 472800T^2 - 372400*T + 144400
$43$	$ T^{12} + \cdots + 281702656 $ T^12 - 5T^11 + 57T^10 - 549T^9 + 5594T^8 + 4579T^7 + 199479T^6 - 704013T^5 + 2466197T^4 + 604628T^3 + 325205264T^2 + 186168128*T + 281702656
$47$	$ T^{12} + 16 T^{11} + \cdots + 398161 $ T^12 + 16T^11 + 242T^10 + 2722T^9 + 27128T^8 + 184562T^7 + 1006843T^6 + 2665897T^5 + 4425128T^4 + 4897922T^3 + 3681137T^2 + 1710641*T + 398161
$53$	$ T^{12} + \cdots + 10736275456 $ T^12 + T^11 + 197T^10 + 537T^9 + 21608T^8 + 4907T^7 + 1544713T^6 + 1081627T^5 + 309849193T^4 + 333866672T^3 + 2378774272T^2 + 7250633216T + 10736275456
$59$	$ T^{12} + 3 T^{11} + \cdots + 17181025 $ T^12 + 3T^11 + 68T^10 + 346T^9 + 2405T^8 + 9436T^7 + 38883T^6 + 103413T^5 + 309051T^4 + 80075T^3 + 95865T^2 + 3316000*T + 17181025
$61$	$ (T^{6} - 4 T^{5} + \cdots - 944)^{2} $ (T^6 - 4T^5 - 195T^4 + 980T^3 + 4205T^2 - 9784*T - 944)^2
$67$	$ (T^{6} - 12 T^{5} + \cdots + 9920)^{2} $ (T^6 - 12T^5 - 124T^4 + 1225T^3 + 3495T^2 - 26200*T + 9920)^2
$71$	$ T^{12} + \cdots + 379314576 $ T^12 + 19T^11 + 361T^10 + 5011T^9 + 72732T^8 + 433561T^7 + 3640131T^6 - 3179197T^5 + 25369279T^4 - 78005442T^3 + 424522728T^2 + 664910640*T + 379314576
$73$	$ T^{12} + \cdots + 5895782656 $ T^12 - 10T^11 + 217T^10 - 106T^9 + 5659T^8 - 86794T^7 + 2397669T^6 - 15928782T^5 + 121341657T^4 - 146528108T^3 + 868311344T^2 + 3749209152*T + 5895782656
$79$	$ T^{12} + \cdots + 12091421521 $ T^12 - 3T^11 + 169T^10 - 1863T^9 + 16402T^8 - 64697T^7 + 1063334T^6 - 2197889T^5 + 9137484T^4 - 59149261T^3 + 899760097T^2 - 2989839590*T + 12091421521
$83$	$ T^{12} + \cdots + 157275696400 $ T^12 + 36T^11 + 807T^10 + 12698T^9 + 169540T^8 + 1761142T^7 + 15679922T^6 + 99772334T^5 + 471260341T^4 + 835919770T^3 - 448874460T^2 - 18016629400*T + 157275696400
$89$	$ T^{12} + \cdots + 304711936 $ T^12 + 15T^11 + 208T^10 + 2473T^9 + 25459T^8 + 134753T^7 + 487426T^6 + 164911T^5 + 2555257T^4 + 44532296T^3 + 369549376T^2 + 238169664*T + 304711936
$97$	$ T^{12} + \cdots + 29659728400 $ T^12 + 14T^11 + 201T^10 - 221T^9 + 8466T^8 - 197655T^7 + 4343155T^6 - 38675505T^5 + 267651365T^4 - 1080221500T^3 + 3783199800T^2 - 10851582200*T + 29659728400
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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)

\(f(q)\)	\(=\)	\( q + ( - \beta_{6} + \beta_{5} - \beta_{3} - 1) q^{2} + \beta_{5} q^{3} + \beta_{3} q^{4} - q^{5} - q^{6} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{7} + \beta_{6} q^{8} + \beta_{6} q^{9}+O(q^{10}) \) q + (-b6 + b5 - b3 - 1) * q^2 + b5 * q^3 + b3 * q^4 - q^5 - q^6 + (-b7 - b4 - b1) * q^7 + b6 * q^8 + b6 * q^9	\( q + ( - \beta_{6} + \beta_{5} - \beta_{3} - 1) q^{2} + \beta_{5} q^{3} + \beta_{3} q^{4} - q^{5} - q^{6} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{7} + \beta_{6} q^{8} + \beta_{6} q^{9} + (\beta_{6} - \beta_{5} + \beta_{3} + 1) q^{10} + (\beta_{8} - \beta_{5} + \beta_{3} + 1) q^{11} + (\beta_{6} - \beta_{5} + \beta_{3} + 1) q^{12} + (\beta_{11} + \beta_{10} - \beta_{9} + \cdots - 1) q^{13}+ \cdots + ( - \beta_{9} + \beta_{6} + \beta_{3} + 1) q^{99}+O(q^{100}) \) q + (-b6 + b5 - b3 - 1) * q^2 + b5 * q^3 + b3 * q^4 - q^5 - q^6 + (-b7 - b4 - b1) * q^7 + b6 * q^8 + b6 * q^9 + (b6 - b5 + b3 + 1) * q^10 + (b8 - b5 + b3 + 1) * q^11 + (b6 - b5 + b3 + 1) * q^12 + (b11 + b10 - b9 + b8 + 2b7 - b6 - b5 + 2b4 + 2b2 + 2b1 - 1) * q^13 + (b7 + b2) * q^14 - b5 * q^15 - b5 * q^16 + (-b11 - b10 + b9 + b7 - 2b5 + 2b3 + b2) * q^17 - b5 * q^18 + (-b10 + b7 - b6 + b5 - b3 - b2 - b1 - 1) * q^19 - b3 * q^20 + (b7 + b2 + b1) * q^21 + (b11 + b5 - b3) * q^22 + (-b11 - 2b10 + 2b9 - b7 - b6 + b5 - b4 - b3 - b2) * q^23 - b3 * q^24 + q^25 + (b9 + b6 + b3 - 2b2 + 2) q^26 - b3 * q^27 + (b7 + b1) * q^28 + (-b10 + b9 - b8 - b7 + b2 + b1) * q^29 + q^30 + (b11 + b10 - 2b9 + 2b8 - b7 + b5 - b2 + 1) * q^31 + q^32 + (-b10 + b3 + 1) * q^33 + (b11 - b9 + b7 + 2b6 + b1 + 2) q^34 + (b7 + b4 + b1) * q^35 + q^36 + (3b6 + 3b3 - 2) * q^37 + (-b8 - b7 - b4 + b3 + b1) * q^38 + (b11 - b6 - b5 + 2b4 + b3) q^39 - b6 * q^40 + (-b10 - b9 + b8 - b7 + b6 - b5) * q^41 + (b7 + b4 + b1) * q^42 + (-b10 - b9 + b8 - b7 + 5b6 - 5b5 + 4b3 - b2 - b1 + 4) q^43 + (-b11 - b10 + b9 - b8 - b6 - 1) * q^44 - b6 * q^45 + (b11 - b10 - b9 - b8 - b6 + b5 + b4 + b2 - 1) * q^46 + (b10 + b8 + 3b7 - 2b6 - b5 + 2b4 + 2b2 + 3b1 - 2) q^47 - b6 * q^48 + (b10 + b8 + 3b7 + 3b6 + b4 + b2 + 3b1 + 3) q^49 + (-b6 + b5 - b3 - 1) * q^50 + (-b11 - b10 + b7 - 2b5 + b4 + 2b3 + b1 + 2) * q^51 + (-b10 - 2b7 - b6 + b5 - 2b3 - 2) * q^52 + (-2b11 - 2b10 + 2b9 + 3b7 + b4 + 3b2) q^53 - b6 * q^54 + (-b8 + b5 - b3 - 1) * q^55 + (b4 + b1) * q^56 + (-b9 - b4 - 2b2 - b1 - 1) q^57 + (-b11 - b10 - b8 + b7 + b4 - b1) * q^58 + (-b10 - b8 + b7 - 2b5 + b4 + b2 + b1) q^59 + (-b6 + b5 - b3 - 1) * q^60 + (b11 - 3b9 + b8 - b6 - 2b4 - b3 - b2 - 2b1 + 1) q^61 + (b11 + b10 + b9 - b7 - b6 + b5 - b3 - b1 - 2) * q^62 + (b4 + b1) * q^63 + (-b6 + b5 - b3 - 1) * q^64 + (-b11 - b10 + b9 - b8 - 2b7 + b6 + b5 - 2b4 - 2b2 - 2b1 + 1) * q^65 + (-b8 + b5 - b3 - 1) * q^66 + (-2b11 + 2b9 - 2b8 - b4 + 2b2 - b1 + 2) * q^67 + (-b11 + b9 - b8 - 2b6 + b4 - 2b3 + b1 - 2) * q^68 + (-2b11 - 2b10 - b8 - b7 + b5 - b4 - 1) * q^69 + (-b7 - b2) * q^70 + (2b11 + 2b10 - 2b9 + 3b7 + 2b6 - 2b5 + b4 + 2b3 + 3b2) * q^71 + (-b6 + b5 - b3 - 1) * q^72 + (b8 - b7 - 4b5 - b4 + 7b3 - 3b1 + 4) q^73 + (5b6 - 5b5 + 2b3 + 2) q^74 + b5 * q^75 + (-b11 + b7 + b6 + 2b4 + b2) q^76 + (b10 + b8 - 2b7 + b6 - b5 - 2b4 - 2b2 - 2b1 + 1) * q^77 + (-b11 - b10 + b9 - b8 - 2b7 + b6 + b5 - 2b4 - 2b2 - 2b1 + 1) * q^78 + (2b11 - 2b7 - 3b6 - b5 - b4 + b3 - 2b2) * q^79 + b5 * q^80 + (-b6 + b5 - b3 - 1) * q^81 + (b11 + b10 - b8 - b5 - b1 + 1) * q^82 + (-2b10 + b9 - b8 + 3b7 - 3b3 - 3) q^83 + (-b7 - b2) * q^84 + (b11 + b10 - b9 - b7 + 2b5 - 2b3 - b2) * q^85 + (b11 + b10 - b8 - b7 - b5 - b4 - 4b3 - b1 + 1) q^86 + (-b11 - b8 + b4 + 2b2 + b1) q^87 + (-b9 + b6 + b3 + 1) * q^88 + (b8 + 2b5 + b3 + 3b1 - 2) * q^89 + b5 * q^90 + (-b10 - b9 + b8 - 5b7 - 5b6 + 5b5 + 2b3 - 3b2 - 3b1 + 2) * q^91 + (-2b11 + b9 - 2b8 + b6 - b4 + b3 - b2 - b1) * q^92 + (2b11 - b9 + b8 - b7 + b6 + b5 - b4 - b1) q^93 + (b11 + b8 + 2b6 + b4 + 2b3 - 2b2 + b1 + 3) q^94 + (b10 - b7 + b6 - b5 + b3 + b2 + b1 + 1) * q^95 + b5 * q^96 + (3b11 + 3b10 + 2b8 + 3b5 - 3b3 + 2b1 - 3) * q^97 + (b11 + b8 - 3b6 + 2b4 - 3b3 - b2 + 2b1 - 3) * q^98 + (-b9 + b6 + b3 + 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	\( 12 q - 3 q^{2} + 3 q^{3} - 3 q^{4} - 12 q^{5} - 12 q^{6} - q^{7} - 3 q^{8} - 3 q^{9} + 3 q^{10} + 6 q^{11} + 3 q^{12} - 6 q^{13} - q^{14} - 3 q^{15} - 3 q^{16} - 13 q^{17} - 3 q^{18} - 10 q^{19} + 3 q^{20} + q^{21} + 6 q^{22} + 8 q^{23} + 3 q^{24} + 12 q^{25} + 14 q^{26} + 3 q^{27} - q^{28} + 7 q^{29} + 12 q^{30} + 16 q^{31} + 12 q^{32} + 9 q^{33} + 17 q^{34} + q^{35} + 12 q^{36} - 42 q^{37} + q^{39} + 3 q^{40} - 3 q^{41} + q^{42} + 5 q^{43} - 9 q^{44} + 3 q^{45} - 2 q^{46} - 16 q^{47} + 3 q^{48} + 28 q^{49} - 3 q^{50} + 13 q^{51} - 6 q^{52} - q^{53} + 3 q^{54} - 6 q^{55} + 4 q^{56} - 20 q^{57} - 3 q^{58} - 3 q^{59} - 3 q^{60} + 8 q^{61} - 14 q^{62} + 4 q^{63} - 3 q^{64} + 6 q^{65} - 6 q^{66} + 24 q^{67} - 8 q^{68} - 8 q^{69} + q^{70} - 19 q^{71} - 3 q^{72} + 10 q^{73} - 12 q^{74} + 3 q^{75} + 6 q^{78} + 3 q^{79} + 3 q^{80} - 3 q^{81} + 7 q^{82} - 36 q^{83} + q^{84} + 13 q^{85} + 20 q^{86} + 8 q^{87} + 6 q^{88} - 15 q^{89} + 3 q^{90} + 51 q^{91} - 12 q^{92} - q^{93} + 24 q^{94} + 10 q^{95} + 3 q^{96} - 14 q^{97} - 12 q^{98} + 6 q^{99}+O(q^{100}) \) 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 + 6 * q^11 + 3 * q^12 - 6 * q^13 - q^14 - 3 * q^15 - 3 * q^16 - 13 * q^17 - 3 * q^18 - 10 * q^19 + 3 * q^20 + q^21 + 6 * q^22 + 8 * q^23 + 3 * q^24 + 12 * q^25 + 14 * q^26 + 3 * q^27 - q^28 + 7 * q^29 + 12 * q^30 + 16 * q^31 + 12 * q^32 + 9 * q^33 + 17 * q^34 + q^35 + 12 * q^36 - 42 * q^37 + q^39 + 3 * q^40 - 3 * q^41 + q^42 + 5 * q^43 - 9 * q^44 + 3 * q^45 - 2 * q^46 - 16 * q^47 + 3 * q^48 + 28 * q^49 - 3 * q^50 + 13 * q^51 - 6 * q^52 - q^53 + 3 * q^54 - 6 * q^55 + 4 * q^56 - 20 * q^57 - 3 * q^58 - 3 * q^59 - 3 * q^60 + 8 * q^61 - 14 * q^62 + 4 * q^63 - 3 * q^64 + 6 * q^65 - 6 * q^66 + 24 * q^67 - 8 * q^68 - 8 * q^69 + q^70 - 19 * q^71 - 3 * q^72 + 10 * q^73 - 12 * q^74 + 3 * q^75 + 6 * q^78 + 3 * q^79 + 3 * q^80 - 3 * q^81 + 7 * q^82 - 36 * q^83 + q^84 + 13 * q^85 + 20 * q^86 + 8 * q^87 + 6 * q^88 - 15 * q^89 + 3 * q^90 + 51 * q^91 - 12 * q^92 - q^93 + 24 * q^94 + 10 * q^95 + 3 * q^96 - 14 * q^97 - 12 * q^98 + 6 * 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	930.2.n.d

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:	\( x^{12} - 2 x^{11} + 13 x^{10} - 9 x^{9} + 60 x^{8} + x^{7} + 263 x^{6} + 823 x^{5} + 1931 x^{4} + \cdots + 400 \) x^12 - 2x^11 + 13x^10 - 9x^9 + 60x^8 + x^7 + 263x^6 + 823x^5 + 1931x^4 + 1170x^3 + 1340x^2 + 1000x + 400
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 5 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( - 17849017864 \nu^{11} + 17625053431 \nu^{10} - 178529954095 \nu^{9} + \cdots + 60463522673300 ) / 107519805038390 \) (-17849017864v^11 + 17625053431v^10 - 178529954095v^9 - 111496966669v^8 - 495858189935v^7 - 1902177059364v^6 + 216123116006v^5 - 22850390835465v^4 - 38781999340004v^3 - 33360822706280v^2 - 14600309174840*v + 60463522673300) / 107519805038390
\(\beta_{3}\)	\(=\)	\( ( 23361760035 \nu^{11} - 106837550849 \nu^{10} + 512829808605 \nu^{9} + \cdots - 15010929677480 ) / 107519805038390 \) (23361760035v^11 - 106837550849v^10 + 512829808605v^9 - 1139133730021v^8 + 2950126373569v^7 - 3751671881315v^6 + 10498141078366v^5 + 3863106684695v^4 + 20833220372194v^3 - 23111477672266v^2 + 148807987112405*v - 15010929677480) / 107519805038390
\(\beta_{4}\)	\(=\)	\( ( - 146654331477 \nu^{11} + 214983815436 \nu^{10} - 1782233600297 \nu^{9} + \cdots - 205909846055800 ) / 215039610076780 \) (-146654331477v^11 + 214983815436v^10 - 1782233600297v^9 + 626624859189v^8 - 9624980905700v^7 - 637596034947v^6 - 48748704399099v^5 - 127545198552457v^4 - 382232489020961v^3 - 302595918088550v^2 - 315224424729940*v - 205909846055800) / 215039610076780
\(\beta_{5}\)	\(=\)	\( ( 259180521843 \nu^{11} - 820941472449 \nu^{10} + 4037834562557 \nu^{9} + \cdots + 53049709415200 ) / 215039610076780 \) (259180521843v^11 - 820941472449v^10 + 4037834562557v^9 - 6329673984486v^8 + 19051303779369v^7 - 17907584046427v^6 + 72430523469356v^5 + 134261867661482v^4 + 278674930944664v^3 - 257444750971091v^2 + 212990793099520*v + 53049709415200) / 215039610076780
\(\beta_{6}\)	\(=\)	\( ( - 604635226733 \nu^{11} + 1137874382010 \nu^{10} - 7789757733805 \nu^{9} + \cdots - 663036463432360 ) / 430079220153560 \) (-604635226733v^11 + 1137874382010v^10 - 7789757733805v^9 + 4727597224217v^8 - 36724101470656v^7 - 2588067986473v^6 - 166627772868235v^5 - 496750299137235v^4 - 1258952186163283v^3 - 862551212637626v^2 - 943654494647340*v - 663036463432360) / 430079220153560
\(\beta_{7}\)	\(=\)	\( ( 302580428763 \nu^{11} - 668487778598 \nu^{10} + 3997049287899 \nu^{9} + \cdots + 103672208737200 ) / 215039610076780 \) (302580428763v^11 - 668487778598v^10 + 3997049287899v^9 - 3500472468789v^8 + 18166764568270v^7 - 4266046224647v^6 + 79043701815307v^5 + 221802656734169v^4 + 560685961527401v^3 + 134311106170100v^2 + 206130812427800*v + 103672208737200) / 215039610076780
\(\beta_{8}\)	\(=\)	\( ( 377873811249 \nu^{11} - 1246157058016 \nu^{10} + 6108799410656 \nu^{9} + \cdots - 228519186690980 ) / 107519805038390 \) (377873811249v^11 - 1246157058016v^10 + 6108799410656v^9 - 10496150942691v^8 + 30636118502149v^7 - 34700982692066v^6 + 115791065278799v^5 + 168010412077976v^4 + 381784730133684v^3 - 359469623547681v^2 + 87053037871525*v - 228519186690980) / 107519805038390
\(\beta_{9}\)	\(=\)	\( ( - 811947191771 \nu^{11} + 1070796378400 \nu^{10} - 8730178441851 \nu^{9} + \cdots - 797530905591380 ) / 215039610076780 \) (-811947191771v^11 + 1070796378400v^10 - 8730178441851v^9 - 1669598790561v^8 - 32876644690520v^7 - 47642147786361v^6 - 156547132073865v^5 - 848523389812191v^4 - 1801907179914651v^3 - 1493427761388610v^2 - 583577757140760*v - 797530905591380) / 215039610076780
\(\beta_{10}\)	\(=\)	\( ( 2196856365657 \nu^{11} - 6036163855492 \nu^{10} + 32819025256117 \nu^{9} + \cdots + 17\!\cdots\!20 ) / 430079220153560 \) (2196856365657v^11 - 6036163855492v^10 + 32819025256117v^9 - 42245615072839v^8 + 156459753762298v^7 - 91084996607083v^6 + 604977651733493v^5 + 1468271773696657v^4 + 3005207185931801v^3 + 578163383466308v^2 + 3110122517771760*v + 1778835879579520) / 430079220153560
\(\beta_{11}\)	\(=\)	\( ( - 2487542474027 \nu^{11} + 6234274597186 \nu^{10} - 33979177074555 \nu^{9} + \cdots - 15\!\cdots\!60 ) / 430079220153560 \) (-2487542474027v^11 + 6234274597186v^10 - 33979177074555v^9 + 35569652132971v^8 - 145106089008796v^7 + 42032352335873v^6 - 567823275403729v^5 - 1830786108310665v^4 - 3452889698168529v^3 - 226203310184226v^2 - 1085086519421180*v - 1579157825751160) / 430079220153560

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} - 4\beta_{5} + \beta _1 - 1 \) -b11 + b9 + b7 - b6 - 4*b5 + b1 - 1
\(\nu^{3}\)	\(=\)	\( \beta_{10} + \beta_{9} - \beta_{8} + 7\beta_{7} + 3\beta_{6} - 3\beta_{5} - \beta_{2} - \beta_1 \) b10 + b9 - b8 + 7b7 + 3b6 - 3*b5 - b2 - b1
\(\nu^{4}\)	\(=\)	\( 11 \beta_{11} + 9 \beta_{10} - 9 \beta_{9} + 12 \beta_{7} + 28 \beta_{6} + 11 \beta_{5} + \cdots + 12 \beta_{2} \) 11b11 + 9b10 - 9b9 + 12b7 + 28b6 + 11b5 + 4b4 - 11b3 + 12*b2
\(\nu^{5}\)	\(=\)	\( 13\beta_{11} - 27\beta_{9} + 13\beta_{8} + 43\beta_{6} + 14\beta_{4} + 43\beta_{3} + 71\beta_{2} + 14\beta _1 + 21 \) 13b11 - 27b9 + 13b8 + 43b6 + 14b4 + 43b3 + 71b2 + 14b1 + 21
\(\nu^{6}\)	\(=\)	\( - 84 \beta_{11} - 84 \beta_{10} + 28 \beta_{8} - 127 \beta_{7} - 114 \beta_{5} - 127 \beta_{4} + \cdots + 114 \) -84b11 - 84b10 + 28b8 - 127b7 - 114b5 - 127b4 + 342b3 - 52b1 + 114
\(\nu^{7}\)	\(=\)	\( - 314 \beta_{11} - 155 \beta_{10} + 314 \beta_{9} - 155 \beta_{8} - 491 \beta_{7} - 499 \beta_{6} + \cdots - 499 \) -314b11 - 155b10 + 314b9 - 155b8 - 491b7 - 499b6 + 109b5 - 665b4 - 665b2 - 491b1 - 499
\(\nu^{8}\)	\(=\)	\( 329 \beta_{10} + 824 \beta_{9} - 824 \beta_{8} - 267 \beta_{7} - 1983 \beta_{6} + 1983 \beta_{5} + \cdots - 3185 \) 329b10 + 824b9 - 824b8 - 267b7 - 1983b6 + 1983b5 - 3185b3 - 1304b2 - 1304*b1 - 3185
\(\nu^{9}\)	\(=\)	\( 3494 \beta_{11} + 1861 \beta_{10} - 1861 \beta_{9} + 2088 \beta_{7} - 93 \beta_{6} + \cdots + 2088 \beta_{2} \) 3494b11 + 1861b10 - 1861b9 + 2088b7 - 93b6 + 5429b5 + 6466b4 - 5429b3 + 2088*b2
\(\nu^{10}\)	\(=\)	\( 8327 \beta_{11} - 12048 \beta_{9} + 8327 \beta_{8} + 12957 \beta_{6} + 13301 \beta_{4} + 12957 \beta_{3} + \cdots + 30924 \) 8327b11 - 12048b9 + 8327b8 + 12957b6 + 13301b4 + 12957b3 + 12779b2 + 13301b1 + 30924
\(\nu^{11}\)	\(=\)	\( - 21106 \beta_{11} - 21106 \beta_{10} + 17022 \beta_{8} - 24483 \beta_{7} - 57656 \beta_{5} + \cdots + 57656 \) -21106b11 - 21106b10 + 17022b8 - 24483b7 - 57656b5 - 24483b4 + 65148b3 + 40117b1 + 57656

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(1\)	\(1\)	\(\beta_{6}\)

\(n\):		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
$7$	\( T^{12} + T^{11} + \cdots + 400 \) T^12 + T^11 - 3T^10 - 17T^9 + 190T^8 - 543T^7 + 1417T^6 - 2116T^5 + 3421T^4 - 3850T^3 + 2960T^2 - 1200T + 400
$11$	\( T^{12} - 6 T^{11} + \cdots + 24025 \) T^12 - 6T^11 + 31T^10 - 71T^9 + 181T^8 - 175T^7 + 1765T^6 - 3260T^5 + 16015T^4 - 13300T^3 + 38200T^2 - 44175*T + 24025
$13$	\( T^{12} + 6 T^{11} + \cdots + 1079521 \) T^12 + 6T^11 + 36T^10 + 214T^9 + 1872T^8 + 4654T^7 + 41001T^6 + 274132T^5 + 1841889T^4 + 6280122T^3 + 14820553T^2 - 2098780*T + 1079521
$17$	\( T^{12} + 13 T^{11} + \cdots + 13830961 \) T^12 + 13T^11 + 131T^10 + 710T^9 + 2850T^8 + 6113T^7 + 24429T^6 + 58058T^5 + 444690T^4 + 915050T^3 + 2765786T^2 + 6813208*T + 13830961
$19$	\( T^{12} + 10 T^{11} + \cdots + 160000 \) T^12 + 10T^11 + 25T^10 - 110T^9 + 2315T^8 + 20450T^7 + 97125T^6 + 443550T^5 + 3576025T^4 + 4917500T^3 + 13006000T^2 - 840000*T + 160000
$23$	\( T^{12} + \cdots + 257442025 \) T^12 - 8T^11 + 99T^10 - 787T^9 + 6706T^8 - 22715T^7 + 149300T^6 - 600535T^5 + 9329740T^4 - 55699925T^3 + 179059575T^2 - 210109275*T + 257442025
$29$	\( T^{12} - 7 T^{11} + \cdots + 74149321 \) T^12 - 7T^11 + 28T^10 - 116T^9 + 2853T^8 - 16794T^7 + 151487T^6 - 925241T^5 + 5424493T^4 - 20556439T^3 + 59446413T^2 - 62498638*T + 74149321
$31$	\( T^{12} + \cdots + 887503681 \) T^12 - 16T^11 + 102T^10 - 357T^9 + 268T^8 + 14988T^7 - 142027T^6 + 464628T^5 + 257548T^4 - 10635387T^3 + 94199142T^2 - 458066416*T + 887503681
$37$	\( (T^{2} + 7 T + 1)^{6} \) (T^2 + 7*T + 1)^6
$41$	\( T^{12} + 3 T^{11} + \cdots + 144400 \) T^12 + 3T^11 - 21T^10 - 133T^9 + 3436T^8 + 5305T^7 + 63515T^6 - 13030T^5 + 233065T^4 - 251450T^3 + 472800T^2 - 372400*T + 144400
$43$	\( T^{12} + \cdots + 281702656 \) T^12 - 5T^11 + 57T^10 - 549T^9 + 5594T^8 + 4579T^7 + 199479T^6 - 704013T^5 + 2466197T^4 + 604628T^3 + 325205264T^2 + 186168128*T + 281702656
$47$	\( T^{12} + 16 T^{11} + \cdots + 398161 \) T^12 + 16T^11 + 242T^10 + 2722T^9 + 27128T^8 + 184562T^7 + 1006843T^6 + 2665897T^5 + 4425128T^4 + 4897922T^3 + 3681137T^2 + 1710641*T + 398161
$53$	\( T^{12} + \cdots + 10736275456 \) T^12 + T^11 + 197T^10 + 537T^9 + 21608T^8 + 4907T^7 + 1544713T^6 + 1081627T^5 + 309849193T^4 + 333866672T^3 + 2378774272T^2 + 7250633216T + 10736275456
$59$	\( T^{12} + 3 T^{11} + \cdots + 17181025 \) T^12 + 3T^11 + 68T^10 + 346T^9 + 2405T^8 + 9436T^7 + 38883T^6 + 103413T^5 + 309051T^4 + 80075T^3 + 95865T^2 + 3316000*T + 17181025
$61$	\( (T^{6} - 4 T^{5} + \cdots - 944)^{2} \) (T^6 - 4T^5 - 195T^4 + 980T^3 + 4205T^2 - 9784*T - 944)^2
$67$	\( (T^{6} - 12 T^{5} + \cdots + 9920)^{2} \) (T^6 - 12T^5 - 124T^4 + 1225T^3 + 3495T^2 - 26200*T + 9920)^2
$71$	\( T^{12} + \cdots + 379314576 \) T^12 + 19T^11 + 361T^10 + 5011T^9 + 72732T^8 + 433561T^7 + 3640131T^6 - 3179197T^5 + 25369279T^4 - 78005442T^3 + 424522728T^2 + 664910640*T + 379314576
$73$	\( T^{12} + \cdots + 5895782656 \) T^12 - 10T^11 + 217T^10 - 106T^9 + 5659T^8 - 86794T^7 + 2397669T^6 - 15928782T^5 + 121341657T^4 - 146528108T^3 + 868311344T^2 + 3749209152*T + 5895782656
$79$	\( T^{12} + \cdots + 12091421521 \) T^12 - 3T^11 + 169T^10 - 1863T^9 + 16402T^8 - 64697T^7 + 1063334T^6 - 2197889T^5 + 9137484T^4 - 59149261T^3 + 899760097T^2 - 2989839590*T + 12091421521
$83$	\( T^{12} + \cdots + 157275696400 \) T^12 + 36T^11 + 807T^10 + 12698T^9 + 169540T^8 + 1761142T^7 + 15679922T^6 + 99772334T^5 + 471260341T^4 + 835919770T^3 - 448874460T^2 - 18016629400*T + 157275696400
$89$	\( T^{12} + \cdots + 304711936 \) T^12 + 15T^11 + 208T^10 + 2473T^9 + 25459T^8 + 134753T^7 + 487426T^6 + 164911T^5 + 2555257T^4 + 44532296T^3 + 369549376T^2 + 238169664*T + 304711936
$97$	\( T^{12} + \cdots + 29659728400 \) T^12 + 14T^11 + 201T^10 - 221T^9 + 8466T^8 - 197655T^7 + 4343155T^6 - 38675505T^5 + 267651365T^4 - 1080221500T^3 + 3783199800T^2 - 10851582200*T + 29659728400
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